Transcript for:
Circle Concepts and Equations

Hello everyone Assalamu Alaikum One Another class in the short revision series Welcome to the chapter we will read today. That is the circle, from the circle chapter. Your college exams or HSC Whatever comes up in the exam today, we I will cover everything I will cover everything There is a condition to watch this class, that is Read the circle chapter at least once before. I don't mind if you forget to stay. I'll show you everything in the formula, butt. At least he needs to be educated, and since Revision class, let's move on a little faster. If you can do the class, you will be removed from the circle. Whatever comes in college exams or HSC Everything will be covered, so let's start. We are starting our first The topic is the ideal of the circle. Equation To see the ideal equation of a circle This is a quadratic equation with three A graph with X and Y properties Quadratic equation This is the first property. The second feature is that here X The coefficient of square and y square is One is the square of X and Y. The equation here is x squared and y. What is the coefficient of both squares? What is the coefficient of both squares? Here are the conditions number one and three. There is no path connecting XY, no intersection. If the equation satisfies these three properties, then The standard of a circle Equation OK, if I give you an example X squared plus squared minus Twice X minus four Y minus one Equals to zero is a circle. Standard equation, look at this. A quadratic equation of XY is x. What is the coefficient of both square and y square? One is not and there is no term that includes XY here. No, it's a circle standard. Equation Well now what we're going to look at is one Conditions for a quadratic equation to be a circle Any quadratic equation containing x and y Any quadratic equation in the world looks like this A quadratic equation in two variables would look like this: Okay, now this quadratic equation is a circle. We will see what the conditions are for this dilemma to occur. The equation will be a circle if A and B are equal. Okay, A and B are equal, three conditions must be met. It takes A and B to be equal to condition number two. Which value cannot be zero? And B is equal to which of the values ​​is zero? It won't work and condition number three is H's. The value zero means that here X Y Will there be any added terms? Or a quadratic equation if these three Characteristics mean these three characteristics mean But that would indicate a circle to explain I can see the difference between this and the previous one. Let me show you the previous equation. One extra condition to comply with the conditions Are you paying attention to ARB's Value One? Square and Y squared coefficient one it is But extra conditions here and next time This condition does not exist, but rather there is a condition that And B must be equal to one, so Don't talk, watch the story, X Square. Plus Y Squared Minus Twice X Minus four times minus one equals two. Zero is what ideal equation of a circle? It was on our previous page, so don't look at it. This equation does not satisfy these conditions. I mean, of course it would indicate a circle. I can now explain this equation with two. Multiply both sides by 2x ^ 2 + 2y ^ 2 - 4x - 8y - 2 = 0 A new equation is not coming. Although nothing new has come, it's just another A version is coming that will indicate the circle. Conditions for a quadratic equation to be a circle What does the condition mean? X squared and Y. The coefficient of the square must be equal to zero. Two two equals zero, not x. This means that there will be no terms containing Y. But three conditions are met. So that doesn't mean it's the equation of a circle. This is the equation of a circle, but it Ideal shape of a circle or not To be X squared and Y squared The coefficient must be equal to one. You understand that it means it is the ideal of a circle. Shape but it is the equation of the circle but Not the ideal shape of any circle now You can take the equation to its ideal form. How can we do this very simple thing called X square? Or y squared divides it by the coefficient Work if you give it. So what is X squared and Y? The coefficient of the square won't be one, you understand? So you can use any equation of the circle you want. Can you bring it to an ideal shape? What is the condition for a quadratic equation to be a circle? The coefficient of B is equal to and not zero for X and Y. There can be no combined positions from here on out. Look, questions come up every now and then. So, notice that a quadratic equation is given. Do a quadratic equation given here. When will the quadratic equation describe the circle? Think about this quadratic equation. What is the condition for indicating a circle in ARB? The values ​​must be equal but neither is zero. It can't be, do you understand the value of ARB? It should be equal but neither is zero. This cannot be not equal to zero. Even if you write it as either or, it is not equal to zero. Even if you write it down, these two are equally valid options. Do it. So none of it is happening towards the last one. Notice that ARB has changed it a bit. If they are equal, what is the value of A by B? You should have told me, "One or Two?" Equals to one and any one not It only happens when it's equal to zero because it's a knot. If one is equal to zero, then the other is not equal. Two will be zero because two equals that means Which is our answer? D The number is correct, just check two things. Correlation coefficient of x squared and y squared Equal to or not, it cannot be zero and is happening There cannot be any terms containing XY. There are no terms containing XY here. Okay, we're going to another number like this. We're going to another number, watch out. Given a quadratic equation Given a quadratic equation, If the equation is a circle, find the value of K. It says, "What is the value of this?" This is who it is. This quadratic equation is a circle. We are in a bit of a complex form, fast. I'll break it, even if you notice. So A plus B plus C is the sum of the squares. The formula is not A plus B plus C. Square's Formula x Square Plus Y squared plus three squared minus Twice X Y minus six Y plus Six 6 Here's how I broke it down a + b + c Everyone knows the formula for the whole square. Surely a ^ 2 + b ^ 2 + c ^ 2 + 2ab + Multiply this by 2bc + 2c. Which is + kxy + kxy - kx + 2y - 2 = 0 Simplify a little more if you want. You can do just two things. It is important to note that a quadratic equation The condition for a circle to be formed is just two conditions x^2 And the coefficient of y^2 is equal and nonzero to xy. There cannot be any term containing x^2 and y^2. Look where it is. So x^2 is here but x^2 is not there anywhere else. If y^2 is here then the coefficients of both are equal and Non-zero yes, accepting the first condition. This time the second condition is what? There cannot be any terms containing xy here xy What are the added terms? There is an XY term here. Here is an XY term. But there can be terms that include XY. No, the term that includes XY here is Pay attention. So if you take the common minus two into This means that this position cannot exist without XY. The coefficient is zero, which means we can't tell who Minus two equals two Zero is the value of K. I realized how much it's worth. There are just two conditions for a quadratic circle to be formed. Things to note: X squared and Y Whether the coefficient of the square is equal and nonzero is x. The coefficient of the term involving Y must also be zero. This means that there can't be any terms containing XY now. If you are a little smart If you were a little smarter, this whole thing would be over. The complex thing doesn't really need to be done. Why are you breaking it? It's not really necessary to break it. You need to pay attention to two things. You have to pay attention to two things. Who is the X square towards? The coefficient of y squared would tell me x. Square Y square where will come from? X You could tell the square was coming from here. Square Y Square will come from here too. From nowhere but X squared Y You understand that their coefficients are equal, not square. Now the second thing is where is X and Y from? You can come to XY when you are the formula. Bhangaba here is a minus to XY. It will come from here and XY will come. If you multiply x by y, If you do, an X Y will appear, and where else? But X will not come from Y, so X What are the words with the letter Y? Tell me. The coefficient of two or more must be zero. From there you can tell what its value is. It's not like I'm trying to simplify two things. You can tell me from here, you understand? Okay, from here, but sometimes math Okay, now we're moving on to the next topic. If an MCQ doesn't come from here, then a Secure A is that from the equation of the circle Equation of a circle with a center radius Finding the radius of a circle from the center The ideal equation is the ideal of two circles. An equation is a way of expressing two things in two ways. Go, this is one, this is one, okay? There are two ideal equations of a circle: Center of the circle The center of this circle is H. Comma It's obvious from looking at it from here. Can you tell me the center is H? What is the radius of the comma? Radius is the radius or radius. And understand If we find the ideal equation of the circle, then this The center is given in the shape from where Finding the radius is very easy, the center being H comma K radius is R radius And if this shape is given, this If it is given in the form, then we have The center or center will be the center or Center The coefficient of x will be divided by minus two. What is the coefficient of Corba X? Tell me twice. G is not twice G divided by minus two. What happens if we subtract the coefficient of GY from GY? What do you get if you divide by minus two? Minus F minus F I realized that everyone was telling me to be a circle. If the ideal shape is given in this shape What is its center minus G minus F x? Divide the coefficient of y by minus two. Divide the coefficient by minus two and the radius or Radius will be the root. Over of G square plus F A square Minus C root over G squared plus F S ^ 2 - c is understood to be a special case. There is this radius, this radius. Notice that we are expressing this with more. I did the radius square or G square. Plus F A squared minus C is this thing. If it becomes zero then we Tell us what the radius will be. The radius does not become zero then it It is called a point circle. Dotted circle Dotted circle or point Circle MCQ comes up occasionally, radius If it is zero, what is it called? And this thing, if it's Greater, If it is zero, what is our radius? Real numbers will come, so we If the radius is greater than zero, then The circle that is found when What is found is called reality. The circle is ours. Real is what is available at that time. Either a real circle or a real circle and this That is g squared plus f squared minus c. If you give this thing a lace, it will be zero root. The negative numbers below mean Will the radius be imaginary then? What will it be called then? Imaginary circle Then it will be called an imaginary circle. Just remember what our radius is then. It will become imaginary or complex then. That would be called an imaginary circle. Imaginary circles are complex numbers. What is imaginary number in the chapter? An MCQ core will come from here. The summary is given in these two formulas. Very important center radius How to get out of here? MCQ is coming, stay 100% sure, see what happens. It says that an equation is given, its center Radius very careful very careful we Math formulas that apply to Do normally in standard form or ideal Apply the formulas to the shape where Applied on top of what is derived. We have to take care of ourselves. Are the formulas based on the ideal shape of a circle? Here is the coefficient of x which is Note the center, dividing it in two. Divide the coefficient of x by negative two to get y. The coefficient of the ideal divided by negative two is On the shape but now what is given here Is there a standard form given in the question? But what will be your first task? How can I bring it to the ideal shape? Divide by x squared and How many 1s does the coefficient of y square make? If you can, divide it by two and see what remains. Tell me x ^ 2 + y^2 + twice x - y. + 2 = 0 Now tell me what will be its center x Divide the coefficient of x by -2. The coefficient of x is -2. Divide the central angle by . What is the coefficient of y? -1 but here you have to take it with the sign y. The coefficient of y is multiplied by the coefficient of minus two. What if our center remains divided? I would say minus one point half minus. I can't find the radius of one point half. Okay, if you want, don't come here for a while. Be smart and pay attention, fast, give two. You have to divide it again by minus two later. If you want to divide by the coefficient of x, Can you divide by minus four? You can understand what I'm saying directly from here. If you can do it, it's great, if you don't understand it, it's difficult. No, do it like this, in standard form. Bring it and then do another circle. The equation is given for its center. Coordinate radius These come in a or Do you understand what comes in MCQs? Both the center and radius will be asked in the MCQ. Actually, sometimes both go, and sometimes not. The only thing I want to do is to make it in the ideal shape. Is there a first step with an ideal shape? What should I do? Divide x by three. Square plus y square divided by three What if three times four is 12 minus four x? Plus three five 15 plus five and minus Two equals two zero divided by three. What is coming now is the ideal shape from here. From you find the center and radius. What will happen to the center? What will happen to the center? Center Our center will be a normal point. Equals doesn't give three dots, something does. Equals to any point if the point In the case, three dots are given in the center. What will be the square of the coefficient of x divided by What is the coefficient of Y minus two crores? Divided by minus How much would it cost? Two comma minus five times two is now the radius. What would be the radius? What is the formula for radius? Radius is route over Of G squared plus F squared minus C G is squaring F, so the sign here is It doesn't matter to us, our center. What is the coordinate of minus G minus? F, please tell me where our center is. What minus G minus F if you want this Can you please put the values ​​of the center in place? So its value is actually minus two. If you replace minus two with plus two, No problem, it's getting better, you understand what I mean. But what will happen in real life? The value of G will be -2 then what is the value of -2 squared and F? The value of minus F is -5/2, so F is Value will be 5/ 2 Notice how much square C is the hole? Looking at the sign, C is minus -2. -2 and the radius since the length is you You must write a single, but if you don't write a single, You'll cut the number now. Calculate it now. Calculate it now. Root over G squared plus F squared Minus C 2/7 2/7 If the equation is given as a unit, then the center I hope everyone can determine the radius. Look at the next one. So check which dotted circle has Is the radius zero the same as the root of the circle? Over of G square one square plus one Square means G square plus F square. Minus C is zero, plus one squared. One square minus one does not equal zero. Do you understand what I'm doing? G squared plus F squared. Minus C is not zero, so it is G square minus plus F square sign I am not considering G Square Plus FA. Square minus C is zero, that means This is a dotted circle and needs to be checked. No, I understand. Okay. Okay, we're moving on to the next topic. The circle chapter is basically divided into two parts. What we have been seeing so far is this. It's very basic, it only takes two chapters. Our circle chapter will basically be One is divided into two parts. 41 is one 42 Both are equally important 41 Here that The topic being discussed is the circle. The equation of the circle is in R42. Tangent This is the equation of the circle. The equation of the circle and here is the equation of the circle Tangent is discussed. Okay, so the major of 41 and the most The important thing is that type. Equation of the circle Determine the equation of the circle a little earlier. What we are doing is giving the equation of a circle. From there, the center radius would be Now I'll do the opposite, make a circle. I will be given various information about Center radius may be given Various other information may have been provided. What we will do is to draw a circle from that information. I'll figure out the equation, a question from here. This type will come, it is very important and It has many categories. There are subtypes, you have different types. The information will be given in a circle. You have to find the equation of that circle, you understand? We were looking at two ideal shapes of circles. So, finding the equation of a circle means What is the unknown in this ideal equation? The numbers are in HK and note the ideal. The values ​​of the unknown quantities in the equation If you can get it out, then the job is done. So, pay attention again, but The ideal shape of the circle is the unknown quantity here. Who is GFC or not me? I would say this is the GFC for a circle. Value is kind of your main target. What if you were asked to find the equation of a circle? Finding these three unknown quantities or these three Either way, for example, I If I tell you the center of a circle Given a radius of two point three Four, then what is the equation of the circle? But you can get it out with a bang. Minus two whole square plus y minus Three-hole square equals four squares. Look, you know the value of HK and the other three. To explain I got the equation of the circle to be this type. In this case, we will assume that what we do is fast. The equation of the circle is either this or this then From the various information given in the question, it Trying to figure out the value of HKR if caught I will do it and if I catch it, the value of GFC will be found out. I'll try to make sure you understand the story. Various properties or information about a circle The equation of the circle from there will be given You will need to find the equation of the circle. Does it mean that in its ideal form the unknown The amounts are in HKR or GFC values. I will quickly take the circle out. The equation is either of these two then Trying to find unknown quantities from data I will do three unknown quantities, so three pieces of information. Normally, it is given in question. Which one do you need now? I think a basic rule is that if you The radius is given by the center or If you are given the radius, you can find the center. Or if the radius is given, then normally What is the ideal shape of a circle that you use? Let's assume that the equation of the circle is this. Find the value of HKR. Normally and in all other cases and in all other cases In the case you assume the ideal of the circle The shape is this GFRC value. Find out now that this is always holding you back. Will it happen or will it have to be caught at any time? You can do whatever you want, but I can't. It's easier if we take it the way I'm saying. Let's start. We begin. Our first type is the center. And how do you get the equation from the radius? Given the center and radius in question The equation of the circle will be found from there. You have to see what a circle says. Find the equation of a circle. Determine the center of which is four commas. Five, aren't you the center of the circle? Having You have reached the center of the circle now. Whenever you find the center of a circle You know what the radius is? Can you come up with an equation for this? The one who applies it So whenever you see the center, What is your brain telling you? Bring me the radius and I'll give you the equation. I'm telling you, isn't that your target now? Finding the radius is given by the center and which passes through the center of another circle Story Do you understand the story? Do you understand what happened to us? Another one? Hold the circle, this is it, this is the second one. The circle is the second circle, take this, ours. The second circle is this yellow circle. Being the second The circle You see, this yellow circle is the second Take this circle and another circle of ours. White Circle This white circle is our first The circle whose equation is mine I need the equation for this white one. The center of the circle is given as four commas. Five is the center of this white circle. How much is it? Four point five I understand now this white The circle passes through the center of the yellow circle. That means the center of the yellow circle is ours. The center of the yellow circle is the white circle. Above the perimeter So, how much is the center of the yellow circle? You could say that looking at the equation, the coefficient of x is minus. Divide by two minus the coefficient of x. Divided by two is minus two y. Dividing its coefficient by minus two gives Three, then the yellow circle We assume that the coordinates of the center C Prime, take whatever you want. No problem, you understand, take whatever you want. There's no problem with our white circle. We took the center. C and I took the center of the yellow circle C. If C is prime, what are the coordinates of prime C? I would say Now tell me this, the equation of the white circle. You need the center, you know what the white circle needs. Do you know the radius of the white circle? The center and any point on the circumference The distance between them is what it is. Radius, find out the radius. Tell me what the radius will be and what the equals are. Two roots over two sides from four If you subtract minus two, you get four plus two. Subtract three from the whole square plus five. Five minus three is square five. What is the square root of minus three? This is me. It means the root of six squared plus two. Square to root 10 Two root ten If the unit is 1, then what is the equation of the circle? What is the equation of a circle? It will be X minus H X minus four holes. Square plus y minus k y minus Five Equals to the radius squared. Finally, our desired answer is coming. You understand that you will be given various information. From there, the equation of the circle is the center. It's very easy here if you give the radius. The center was given, the radius was given directly. It wasn't given but I took it out. Write a little Bengali, you understand. How to write Okay, let's move on to the next number. So what is this? It says that a circle is given by this circle. Note that the equation of diameter has four options. So is the diameter of a circle a chord? A chord that passes through the center. What passes through the center doesn't pass through me. Now, the circle that is given is What is the center of this circle? What is the center of this circle? X Divide the coefficient of x by minus two. Divide the coefficient by minus two to get Y. Divide the coefficient by minus two to get x. Divide the coefficient by minus two to get the coefficient of y. Division by minus two is fine now which Diameter The equation will be satisfied by this point. Not me. You said to boil it, six Y's instead of X's. Instead of minus three, it's either six or minus three. And minus two equals six plus three into two. Minus two equals zero, that means This is ours. If no answer options are given, it is Why couldn't it be the diameter of a circle? So many, which one will you find the equation for? No, we are fine. Now let's go into some concepts, some circles. In the concept, we see a fast Notice that the circle has given its ideal shape. We always follow the ideal formulas. When you derive on the shape Applying the formula is definitely ideal. It has to be applied on top of the shape. The ideal shape of a circle is this circle with the x axis. The point where it intersects AB means that What is the length of AB? It is two roots over d squared minus. If C intersects, then he may not intersect. If not, I'll see what the story is later. Can you touch it or is it too high? He will not even touch or cut these later. I'll come back later. Okay, now this circle is Y. How much will it intersect the axis? The length is this. You understand that if it doesn't intersect, then its value is Either zero or an imaginary number How will the complex number come to him? You two are not showing any evidence. Get the coordinates of the points and then add them. The distance between them can be found. Okay, now. Our next story is about a circle. The condition for touching a circle is the axis. The condition for touching a circle is if x The story won't be like this by touching the axis. Tell me what it will look like, details are coming. Coming later, you said that if you touch the x-axis This is how much he will subtract from the x-axis. So how much does a circle intersect from the x-axis? So if it touches the x-axis If you do, don't look at this cut part as zero. How much did he cut off from the x-axis? Just touch a point like a dot It means the intercept from the x-axis. Zero from there we can say yes Square minus c equals zero there. From this we can say that G squared equals C. So if a circle touches the x-axis A circle touches the x-axis. If G squared equals C or vice versa How does a circle touch the x-axis? What is the condition for D squared equals C? Now you can tell me from here. Tell me if it touches the x-axis. Is this the radius of a circle? If it touches the x-axis by this much, No radius So, so, a theorem in small classes Are you reading? If the tangent point and the center connect Then the line segment you get is tangent. On Perpendicular to the x-axis but a tangent Perpendicular to the tangent means it is Y. Parallel to the axis means that Notice the length. Is this ours? Isn't the center's worth crores? So that means if it touches the x-axis Absolute value of the centroid of the radius equal to the center The radius is never negative, but The center's crore positives could turn negative Maybe if it's on the top, the center's crores If the positive is on the downside, the center's crores It would be negative, but the radius is positive. It is not negative, so it touches the x-axis. What is the condition if G squared is equal to C or The radius is equal to the absolute value of the centroid of the center. Modulus of minus F is right at the center. How many minuses are the coordinates of the center of a crore? You understand, you have to take the absolute value of G minus F. You understand, you understand, if you told me the same way, Intersecting from the y-axis by touching the axis Part Zero to Root Over of F S Square Minus C zero means F A squared. Equals to C touching the y-axis. If two are equal or this is equal, then y. What is the radius touching the axis of the center? The absolute value of the angle is equal to the value of the angle. Note that If the radius isn't this small, tell me this much. The radius is the absolute value of the central angle. Now if it touches both axes So the value of D squared will also be C F. The value of square will be C or G square. Equals F S Square Equals to C R Notice that we The radius is the radius we have. Both the central axis or the crore are of absolute value. Since equal radius is not negative Shall we take the absolute value and another small one? Concept tangent point coordinate X If this point touches the axis Tell me what the coordinates of the x-axis will be. That means crores but zero is its opposite. The food will be the same, so don't tell me it's completely common. Sense minus c comma minus g comma zero y If it touches the axis, the angle will be zero in crores. Will be Tell me how much is a crore, zero comma minus F? If two axes touch, two You must have understood the touch point. What will happen to minus g comma zero? What will happen to this? Zero comma minus Just keep these concepts in mind. It's very easy to take, nothing to worry about. A small number I see a circle is in perfect shape. Always make sure the formula is in perfect shape. If the equation above is applied, then it is never ideal. Not in shape, but in ideal shape You have to go and then do it in the ideal shape. There is no problem, this is the X axis. The amount of the segment that intersects the y-axis You determine that the value of GFC. Take it out and tell me what the value of G is. What is the value of three F's minus three C's? What is the value minus 31 then you are in Bengali? Let's write this as the intercept of the x-axis. The length of the segment intersected from the x axis. I am not writing the Bengali version of the part length. I expressed the x axis with LX. Length of the intersected part from to root over of What is the square root of G minus C? Of three square minus minus 31 How much is it to root? Four four into 10 root four is taken out Two times two will make four, right? Four. Route 10 Single session Intersecting from the y-axis in the same way Length of the segment to root over of F squared Minus C is the intercept from the Y axis. Find the length. What is the value of 2 into F? Minus three, you understand, won't make any difference. You are squaring the difference between plus and minus. The four routes will not be the same. Ten Single clear is also available in MCQs. Sometimes they are in MCQs or How much? Look what the next one says, a question as simple as water. One thing to remember is the formula for a circle. If you forget, a beautiful picture If you draw any number in the circle, Math becomes much easier when you draw pictures. The possibility of making mistakes is reduced, not the image. If you can draw, it would be great, very great, but If you have any problems, please send me a small picture. You can take it in shape or you can rough it up. Rafe, can you draw a circle? The equation is given. This circle touches the x-axis. The value of C is the coordinate x of the tangent point. If Akshay touched it, who would say yes here? What is the value of minus two on the x-axis? Touching means question by question According to you, write it touching the x-axis. If so, does G square equal C not mean What is the value of C minus its whole square? What is the value of plus C? Right then what does the touch point say? If the coordinate touches the x-axis What were we looking at the coordinates of the touch point? So you can picture it in your head. Touching the x-axis will be visualized. What is the coordinate of the tangent point if G comma zero is the coordinate of the center. But since crores are on the x-axis What does zero crore zeros mean? Two comma zero dots equals equals. No, normally three marks are equivalent. Do you understand the equivalent of the last bit of Bengali language? Write the x-axis at the point x. Write down the coordinates of the tangent points. Okay, now we have some more concepts. I'll see, just give him a little attention. You have to pay attention to a circle. The standard equation of this circle is How much is the center for me? I would say minus G minus F minus G. Minus F, what is written above? Center Located on the X axis or Y axis If the center is on the x-axis Either tell me the center is on the x-axis. If located, the value of F will be zero. No So the center is on the x-axis. If the value of F is zero, these are normal. You will need it later in the numbers. This will allow you to find the equation of a circle. The type that comes up the most in the exam is What does finding the equation of a circle mean to me? What does it mean to find the equation of a circle? Here are three ways to calculate the value of GFC. Finding the unknown quantity is one of the three GFC questions. Give me the condition that there will be value of GFC from them. To find out, there are three conditions that can be like this. The center is on the x-axis, so what is a Are there any conditions given? See if there is even one known zodiac sign. The value of F is zero. Two conditions may be given in the question: Example: Coordinates of two more points The ball is given with two more dots. Then you go. I'll see if I can take it easy and see what happens next. If the center is located on the Y axis Then you would say, "You don't know the number of zero crores." That means the center of a circle is Y. What is the value of g when it is located on the axis? The zero center is located on the Y axis. If the value of G is zero, then we are next. Going to the number, a circle is the root. Does pointwise mean the equation of anything? Think circle or think straight line. All the dots above it are The points will not solve his equation. The equation of this circle is given by This is a point on this circle. The main point is that point. Could you tell me about this circle? The one point above is the main point now. Any point above it has its equation It will solve the main point, since That's the main point, the main point, the main point. What is zero Y in place of X? Zero in place will always become zero, right? Just tell me everything will be zero. There will be a C left, so what do we do from there? I can write C equals two. Zero means a circle that is essentially a point. If C equals two Zero Clear is more fun. Two stories on the X axis or the Y axis Touching the main point You are doing this here but on two conditions. Is the X axis or the Y axis the origin? Touch the point I said two conditions to the x-axis: The circle touches the x-axis at the origin. Two conditions apply: it does not touch the x-axis. Doing it and passing the main point So there are two conditions, not two unknowns. The value of the zodiac sign will be clear. Okay, let's see if it touches the x-axis. Make the x-axis the origin at the origin point. The point touching the x-axis Touching means G square. Equals to me. You said the condition is to touch the x-axis. This is again going through the main point because what Touching the x-axis at the origin He must have eyes that are directed towards the main point. It is understood that if we pass through the origin, C Don't you see that the value is zero? Where is it coming from touching the x-axis? Where is the x-axis coming from? Touching the point with the original point The values ​​of two unknown quantities are being found. The value of G has also gone. The value of F is zero, the value of C is also zero. So you would tell me three unknown numbers, one To find the equation of a circle, you need these three If you need to find the value of an unknown quantity, Given three conditions, find the value of all three. There will be two conditions here, two values. Another thing has come out of here. Can you tell me where the center will be? You said where the center would be, right on the Y axis? Know the number zero comma minus F. Or in the same way, if the y-axis is the original The y-axis touches the origin at the point Touching the point F Square F S Square Equals C Equals to Zero Clear So where will the center be on the x-axis? Zero Comma minus G comma zero this was ours The concepts were our concepts. Here are some summaries if you want them. You can take a look, but you don't need these, huh? You can take a look if you want. Okay, okay. Coscenta Read this, it means that in this chapter 41, basically Two types of numbers. Are both types the same? The equation of the circle will be given by the equation He will ask you to find out various information. He will tell you what information is. How far is the center radius from the x-axis? How much of it is cut off from the Y axis? The area of ​​Habijabi is the first thing that exists. The equation will be given in type from there. Different types of information detection The second type is different information. I don't see the equation from there, given. Given the center and radius, the equation would be I'm just trying to figure out what a circle is. The center has provided some information and that With the cardinal point So he's asking me to find his equation. The first one What information is given to solve the equation? It says then that circle has two axes again. To find out what part intersects from They say it's a combination of two types. You understand, but here are two numbers to match. First of all, this much has been done. Given is the equation of the point passing through the origin. If it were a different number, it would have been two. Okay, another circle for the number. From the equation, that is the x-axis and the y-axis. I don't see how much of it is cut off from the One digit was coming up, two digits were coming up, combine them. We make it number four. I will quickly find the equation of the circle with the center. Given I'll write a little bit of a hazy line at the center of the circle. Write these coordinates down nicely. If you give me four point minus five, then Tell me the value of G. How much is the value of minus four F? How much is five? Before that, you will take our circle. The equation How much before we assume that our circle The equation is x squared plus Y square plus twice G x plus Twice FY plus C equals zero. My target is to find out the value of GFC. My target is to achieve the value of GFC. Okay, let's find out the value of GRF. I'm already taking out the center. It means that the value of GF has always been out. Remember, being centered means being GF. The value of C has been found out. Will it be needed? How to do value, the origin goes through the origin point passes through the point so C = 0 Then the equation of our circle is The equation of our circle is found to be The value of GFC has been revealed. You see, there were three conditions given. What do the two coordinates of the center mean? From three conditions to one point Our circle equation is out. X The equation of our circle will be x. Square plus Y square plus Twice G X Minus 8x Plus Twice FY plus zero equals zero. Then how much does the two axes intersect? In Bengali, we will write the intersection of the x-axis Part Tomorrow will be Baisakhi, it seems, the X axis The intersection of the two roots over G is The value of square minus C has already been found. Minus four is its whole square minus zero. So how much is four times twice? Eight eight What is the intercept of the unit then from the y-axis? Two root over of F squared minus C F What is its value? Five minus C is how much? I said 10 I understood everyone. Okay, let's go back to the previous type. Determining the equation of a circle Three unknowns in the equation These words about GFC are important. GFC or HK and whatever you think is yours The target will be the value of these three chapters. Three conditions are required to find out the question. The values ​​of these three will be given from those. We need to find out who belongs to this category now. We will see many subtypes, our first sub Type tangent to the X or Y axis Notice the circle, the first subtype is X. Or circle tangent to the y-axis. What is the given number? One comma two. A circle with center on the x-axis From the y-axis with its equation touching Find out how much part is cut off. The number is the number but the coordinates of the center Given that the value of GF has been revealed But what does the center mean minus G minus? If not, then the value of GF has been revealed. It will take a condition for that. G square touching the x axis You haven't caught Equals to C, what should we do? Let's say our equation of the circle is The equation of the circle is square plus y. Square Plus Twice G X Plus Twice FY plus two C Standard form we use it in all cases I will only take one field if If the radius is given, then it does not hold. That's HK and X minus H. Square plus y minus k is the whole square. I'll take equals and square and do the rest. In this case, I will assume that it will be an advantage. Okay, the coordinates of the center are given. What is the coordinate of its center minus G? How much is minus F? Give me that. You would say that it is one comma two from here. Its value is the value of minus one F. Minus two touches the x-axis. Notice that it touches the x-axis. It means D squared equals C. That means D squared equals C. From there we have I can say that the value of C is The value of One C is One. So the value of GFC has been found out. Meaning What is the equation of the circle? It said x squared plus y squared plus Twice G X Plus Twice G X Plus Twice F Y plus C equals zero C. Value is being What does One First Part say after the end? How far from the axis does it intersect the y-axis? Tell me how many parts will be cut off by the root. Over F squared minus C Two root over two root over two square The value of F is two minus one. Root three is written as a unit of length. This will be correct since the length must be written in units. There is Okay, next one, next one, pay attention. Says the same type of circle x axis What was I saying by touching a circle? Finding the equation of a circle with an unknown quantity What does it mean that all the unknown quantities in his equation He finds out the value of those and draws a circle. Equation unknowns three GFC three Three conditions are required for value: Question Read a line touching the x-axis. Condition one condition then these two dots It means two more conditions. Given, notice the x-axis. Touching and with two more dots What do we assume about the equation of the circle? No, the equation of our circle is x. Square plus Y Square plus twice G X plus twice FY Y plus C Equals to zero now since he is x Touching the axis means G square. Equals to C, isn't that what he used to tell me? Since D is touching the x-axis Square equals to C then we have How does the equation look like? Our equation It stands for x squared plus y. Square Plus Twice G X Plus What is the replacement for Twice FY Plus C? I'll write G square. Do you notice one? Applying the condition reduces an unknown quantity. Didn't it go? Pay attention to the three unknowns above. The zodiac sign was as follows, notice how many unknowns there are. You are not a two-digit number, you are a GRF. If you can figure out the value, it's a sure thing. Don't you understand? Okay, now it's GRF. How do I find the value? Two more conditions. This circle is given as one comma two. If this circle towards the point is one comma two If it is a comma, then one comma two. The coordinates of the point are the equation of this circle. Will you understand me better? There are two conditions given, which are: The circle is two more. Two more points to the point, one point to the point. If comma two is followed by a period, it is one comma two. The point will satisfy the equation of this circle. One will be boiled. Square plus two Square plus twice G Plus Four F Twice G plus four F plus G square Equals to zero, pay attention to us again. Does the circle go through three commas and two dots? That means the coordinates of three commas two dots. What will make this equation complete for us? Since the circle is three commas and two dots, This is the coordinate of three commas two points. This will also complete our equation. Equals two six G plus four F plus G squared equals zero. This is Our second equation Okay, okay, look at the two equations. Two unknown zodiac signs. Who are our unknown zodiac signs? Who is DRF? Two equations, two unknowns. How is the work finished if you solve these two zodiac signs? Solve by subtracting one from the other. Come on, aren't you reading the adoption process? Number equation to two number equation Or what happens if you subtract one from two? They used to tell me from one square to three squares. Subtract means minus eight minus one. Subtracting two squares from two squares What would be left if we removed six Gs from two Gs? Minus four Gs minus four Gs then four. F to four F minus G squared to G Square minus equals zero, so here From 4g = 2 minus 8 then what is the value of G? Now I understand that G equals two minus two. How do I find the value of F? The value of G? Give me any one of them and I'll finish the job. If you set the value to one number, then G. If you set the value to one number, Look, the value of G is everywhere. You will only be left with the value of F. Sit down and see if the value of F will be minus. Five by four, okay, I'll do a little calculation. Trying to do one square plus two squares. How many five plus two G means? Two into two minus two plus four F plus G Square minus two equals its whole square. The value of F starts from two zeros. I realized that five plus two would come in. Minus two plus four x plus two squared Equals to zero If you solve this, which If you get minus five by four, it's GRF. Just plug the value into our equation. I'm not giving away the rest, okay? Where should I put these two RF values? If you sit here, then our circle will be complete. The equation will come out. You see, there are three conditions. If you know all three, you will get out of three conditions. What was this point touching the x-axis? Two conditions are given by this point. There are three conditions, now these three conditions come true. These three conditions can be given in various ways. Actually, it could have been given in various ways. The next one is another subtype X or Y. The axis at a specific point Touch the X or Y axis to make a Touch at a certain point, already two There is a condition, please state the question in the middle of this statement. Find the equation of the circle such that x Touches the axis at point four point zero. This is the first one to touch the x-axis. The word is a condition at four point zero. This is the second one by touch. I understand the condition, then what does the Y axis say? From six units long one takes a jatt There are three conditions given for touching the x-axis. Touches the point four point zero A chord six units long from the y-axis. If we cut it, we won't hold our circle. Equation The equation of our circle is x. Square plus Y Square plus twice G X plus twice FY Y plus C I think I understand equals to zero. Touches the x axis. Touch the x-axis at point four point zero. How many touch points do you remember? Minus G comma zero No, this is the touch point. The coordinates of the point of contact are given. Four comma zero four comma zero then me Tell me the value of G. What is the value of G minus four one? I have found the value of the item, what do I need now? The value of FRC touches the x-axis. It means D squared equals C. If it touches the x-axis, then D squared = From there to C, how much is the value of C minus? Four square or 16 or 16 GC is out, do I need F? Does F need a value? How do I find the value of value F? Given a line six units long from the y-axis Jack cuts it means the y axis How many sixes are there in the fractional part from the number y? What is the number of sides of the segment from the axis? There is a circle segmented which means how much it is segmented. How many parts are cut off from the Y axis? Root over F squared minus C is not it. The value of six means F square minus. C equals two. What is the number of three squares? That means our FA square Equals to 9 plus 16 is 25 then F's What will be the value plus or minus? The value of Five GFC has been revealed. If its value is two, then the circle is actually How many will come? Two circles. Actually, two will come. X squared plus Y Square plus twice G x plus twice G x means minus 8x plus twice. FYI this is the plus minus here but With two circles plus, you get a minus. One plus three equals two. Zero, everyone understands, you can draw a picture if you want. Can you do this math by drawing a picture? It is very clear that the two The equation will come, notice if I draw a picture. You draw the fast X and Y axes. You throw the axis in the air, fast X axis. And put the y-axis in the Akaye, this is Our y-axis and this is our X axis Okay, touch the x-axis at a point. Pay attention, hold our circle. Yellow Lower the circle a little. Okay, then this. You have to go down a little further, go down even lower. It's not fitting, wait, wait. Wait, this is a circle, this circle. At what point does it touch the x-axis? Four point zero is touching the point. and six units long from the y-axis It means that a chord has been cut. How long is the six? Just think about it. Exactly one more circle can be found that x The axis touches the point four point zero. Will do So No This is not being selected. Why are you thinking about the bottom? Could you find another circle like this? It will go like this. Wait a minute. Wait, where is the circle? The circle has vanished. Look, I'm here now. I'm giving you two circles that you can find. Both of them have the x-axis at four point zero. will touch the points and and the y-axis Surely you could understand from that, everyone and What is six units long from the y-axis? Cut the chord If the circle is too big, it will be too big. So, if you want, you can do it with just a picture. This can be done by applying the Pythagorean theorem. Many of you are trying to try it yourself on the board. Actually, if you do this much Okay, everyone, I understand. Next time. Type touches both axes. Given two conditions but one circle both Read the question by touching the axis. Given two circles, tell me. Find the equation that touches both axes. You do and you go through one comma. Tell me if a circle has two axes Touching If two axes touch, then he is Will be defined in a specific quadrant Either it will be in the first quadrant or it will be here. Either here or there is common sense. If a circle touches two axes, it The first and second quadrants never coincide. Will be Don't you understand what I'm saying? There is no such circle. Two are in the quadrant, but two If a circle touches the axis It must touch any two axes. There will be a definition within a quadrant. Notice that this circle is divided by two axes. He is touching the first quadrant. I understood that meant he was the first. Will pass through all the points of the quadrant. Read the question again, circle one or two. Touches the axis and one point eight One point eight goes through the dot. Point Any point in the first quadrant If you told me no, it means that we In which quadrant will the center of the circle be? In the first quadrant it is me fixed it is me I know, I understand, if the center The first quadrant contains its sides and its crores. What will happen to both Bhuj and Crore? Tell me the sign will be the same. Or will both of them be the same? It's just common sense. The signs of both will be the same, both are positive. So there's no chance of both being negative. No, both are positive. Here, both are positive. So what is the equation of our circle? Will be Say X square. Square plus Y square plus Twice G X Plus Twice FY Plus C it was supposed to be but Since two axes touch two What is the radius when it touches the axis? The absolute value of the central denominator is not equal to The sign of the two sides of the triangle will be the same, which means The story won't be like this, plus twice. G X plus twice G Y plus G Square Equals Zero So there is only one unknown quantity, is it? How can you find the value of G? Draw this circle one point eight point. This circle is one point eight. A dot means one point eight. The coordinates of the point are the equation of this circle. It will boil down to plus twice G plus eight. Twice 16 g plus g squared equals two. If it's zero, tell me. Square Plus 18 g plus 65 do it or use a calculator If you do it, it's either five, 13 or 65. You understand. Square Plus 13G Plus FiveG Plus 65 equals zero in the middle term. Which turns G Plus Five into G Plus 13 equals zero from which we can say The value of Pari G is either minus five or Minus 13 so we have two circles like this. It is possible to get We get two such circles. possible that touch two axes and One point eight goes to their place. Tell me what their equation is. X Square plus Y Square plus twice G X minus ten X minus 10 Y Plus 25 equals zero. One is x squared plus y squared plus Twice G X Minus 26 X Minus 26 Y Plus 169 equals to zero, you understand, two circles You can find one that looks like another. It will be huge, but it will be on two axes. Will touch And what do you understand by the one point eight point You can do it if you apply a little common sense. Notice that huge circle. Did I draw that and touch the two axes? What is the coordinate of the center? The radius of five 13 point 13 one will be five. What is the radius of the other one? 13 Everyone, of course. You understand, right? Okay, the next number. Just look at what he's saying. Find the equations of two circles which What does X = two zeros mean on the Y axis? This is the y-axis, y = 0, this is x. The axis and the line x = a touch each other. Do you understand the story? Is this ours? Y equals two Zero, watch out, this is our Y. Equals to zero line or x axis it is This is our x equals zero line. Or the y axis and this is x equals The line to a touches these three lines. It is possible to find two such circles. One on top, one on bottom, very common. Doesn't that make sense? If it was X, you would have told me. How much distance will this be? Is it really that much? What Is half the diameter the same as the radius? Half the radius is then A by A by. Two up, one down. You just have to draw the picture on the next page. You can see it's done in a little detail. Take it, you guys see, okay, next number. I'm going to see what the next subtype says. Three-point circle Three-point circle How many unknowns are there in the equation of a circle? Zodiac Three points of attention, three conditions are required, three Passing through the dot means giving three conditions. There is a question, read it, minus one circle. Three comma four minus one comma four four comma Three conditions crossed by two dots But the equation of the circle is given. Let's assume that the equation of our circle is x. Square plus Y Square plus Twice G X plus twice FY Y plus C Equals to zero, that's how many unknowns there are. Find the value of the three GFC numbers. Three conditions are needed to do this or three Equation Three equations are given, these three If you see a dot, pay attention. Minus three comma four goes through the point. What does minus three point four mean? This will solve the equation minus three. Square plus four Square plus two G Into minus three plus twice F Into Four plus three equals zero then Minus one point four semicolon minus One point four if the dot is minus One point four points is the solution to this equation. Will you solve this equation? Solve it. Minus one square plus four square Minus Twice G Plus 8f plus c equals zero then then Four commas to four commas This point goes through this. You will solve the coordinates. Solve the four. Square Plus two square plus 8g plus four f plus c equals two Zero three equations three unknowns The work is done once you solve the equation. Three equations. If you want to use a calculator to find three unknown quantities, It can be done, but only if you use a calculator. Full marks will not be given to the academy. Will give No, you are not a replacement or replacement method. It's very easy to do one thing and then another. You can do it by going to the details on the next page. Just solve the equation and see if it's done. Look at how to do it, number one. From two numbers three equation three Once you solve the unknown quantity, the work is done. I subtracted two, I subtracted three from two. The value of GFC has been revealed, okay? Okay, we're going to the next number. We're going to the next one. In numbers I'm going here, tell me which line is the center. The center line will be on top. Read the question. Twice x minus y equals two. A circle centered on the three lines. The circle is given a condition at this point and The equation of the circle passing through this point is found. Do three conditions, there is a circle there. Two points lead to two conditions. Already right and its center and its The center is above this line, which means three. Are there any conditions? Tell me three conditions. But right, this is one condition, these are two conditions. We assume that the equation of the circle we are We assume that the equation of our circle is It's like this x Square plus Y Square plus twice G x plus twice FY plus C equals zero. I need to find the value of target GFC. Finding the value of the target GFC is How much is the center for me? I would say its center is minus G minus. F is the center of it minus G minus F. The center lies on this line, which means The coordinates of the center are the equation of this line. It means the coordinates of the center will be If you solve the equation of this line, Boil minus twice G plus F = 3 = Two three is number one. Equation G FC Three unknowns Three If you can figure out the equation, you've got one. The equation is out, but the remaining two are out. How to find the equation? These two points will be on the circle. Dot means these two dots. Solve the equation of the coordinate circle. Three Comma minus two is this point on our circle. You wouldn't solve the equation, you would tell me three. Square plus minus two square plus Six G Six G Minus Four F Minus Four F plus C equals zero again. Minus two comma zero, minus two comma zero, this is it. Since the point is also a point on the circle, Its coordinates are the equation of our circle. It's okay to cook, tell me minus four. G plus zero plus C equals zero. This is equation number two. This is equation number three. Equation GFC Three unknowns Three The equation is coming, solve it and the job is done. It can also be done with a calculator, but When you return, you have to do it manually. Could the details solution be on the next page? Okay, let's move on to the next number. We are going to the next number. I'm going to read the question. Read the question. Give this a star, the previous ones also have stars. All of them are a little too starry. More stars, the numbers I'm doing today All of these are very important. A test with such a radius will come. Circle one comma one dot and the center of the circle is this What are the three conditions on the line? Yes, tell me the radius is given. A condition passes through this point. Two conditions and the center is on that line. The three conditions located mean that I am the equation I can figure out the ideal of a circle quickly. The equation must be assumed to be the equation of the circle. This is what I was telling you if the radius If given, then HKR is included. I will assume the ideal equation in all other cases. In all other cases, GFC is here. The radius is given, so why don't you tell me? Here the radius is given as radius. How many roots of ten by two can I write in Bengali? The radius is this much. So what will be our equation for unit x? Minus H Hole Square Plus Y minus K whole square equals Two and square Wow x minus h hole square plus Y minus k whole square equals two. What is the square of the radius? Ten by Is HRK's target four ten by four? Finding the value HRK is two equations. Let's go to two chapters, we need two equations. There are two conditions given, one comma one. Crosses through the point and the center What is the center of this line? Everyone used to tell me how big this center is. The center is H comma. Who is the center since it is on that line? Located Y Equals Three X Minus seven means the center. The coordinates of the center of this line are Who will solve the equation? Equals to three H minus seven K. Equals To Are you watching 3H minus 7? But there is an equation related to An equation regarding HK is coming but Another one is coming, how about another one? Or our circle is one comma one. Bindugami means one comma one. Set the coordinates that will satisfy this equation. So one minus h is whole square plus one. Minus K Equals to ten by Four is now equal to three minus H. Replace seven with this one minus one. H hole square plus one minus three H Plus Seven equals 10 by four. Square is okay I just want value Who is setting it? Value K equals three H minus Seven, what should I do now? Simplify it. You understand, just simplify this thing. What remains after simplifying the equation is That is four H squared minus 20 H. Plus plus 25 equals zero just The formula for the square of A minus B is Twice H will come from there, if things break. Minus five is its whole square equals two. Zero, I have found the value of H. I got the value of H by five. To Because this value of H is placed at number one. If so, who will get the value of it? H If you put this value here, What is K = Two Three H Minus Seven or K? You will get its value in half a year. Last straw What does everyone understand in the end? I got it. Tell me the coordinates of the center, radius. You know the work is done x minus five. Two is the whole square plus y minus k. Its whole square equals 10 by four. Clear, then? Okay, we're on to the next number. I am going to the diameter of a circle. If the endpoint is given, from there How do we find the equation of the circle from there? Let's say I have a circle. What does it have? This circle is the endpoint of the diameter. Take one diameter of the circle and this is its Given two endpoints, assume that The two endpoints are X-One. Y One X One Y One Another Take the endpoint X to Y. Now, tell me basic common sense. From you this circle without any formula But can we find the equation of diameter? They are given two endpoints. The middle point is not the center, tell me. Can you get the coordinates of the center of the diameter? The length is half its radius, the center If you know the radius, the equation becomes another one. A beautiful equation comes with another beautiful formula. There are two of them in a circle. The endpoints of the diameter are two of the diameter. If the endpoints are x1 y1 and x2 y2 then The equation will be x - x1 * x - x2 + y - y1 * y - y2 = 0 It is very important to remember this. An important formula for a circle If the two endpoints of a diameter are X one Y one and X two Y two Either the equation of that circle will be this, we Let's go through some examples. Our first Look at the example. Says one Look at the circle, it says these two upside down. The line connecting these two points of the point The circle drawn along the diameter It means two in diameter. Equation of a circle given its endpoints And the circle is intersected by the y-axis. Part number four, but this is the first part. Two next part two then circle Tell me what the equation will be. Tell me. X minus X one diameter If the endpoint is given, then Is the equation of the circle x minus x? One into X minus X two plus Y Minus Y One plus Y minus Y one into Y minus Y equals zero. Just put the values ​​in and give them X minus. X One X Plus Four X Plus Four Into X minus X Two plus Y minus Y One y minus y one into y - y2 = 0 Multiply it and then take the form From there, the segment from the Y axis Find out the details on the next page. The solution is given, take a look for yourself. I can do it myself. Actually, by yourself. Can you start the next number with four? CQ MCQ comes everywhere CQ MCQs come everywhere, see what they say. A circle passes through the origin first. Condition of both X axis and Y axis Three three and five respectively in positive terms The equation of the circle is divided by three. But the conditions are given with the main point. Goes three from the positive side of the x-axis. The part cuts from the positive side of the y-axis. Cut it into five parts, then draw a little picture. You are the maximum number in a circle, not a figure. I can draw as many numbers as I want today. I'm showing you that you haven't drawn the maximum picture. We can do almost everything with pictures. Okay, we'll see everything in the picture at admission time. The X axis and Y axis can be done with Okay, we have a circle like this. The circle that passes through the origin X Three parts cut from the axis Y from the axis The circle will look like this after being cut into five parts: Tell me, isn't that the main point? Going from zero comma zero to the x axis x How much do three positive x's intersect the axis? You notice two positive aspects from the axis. From the positive axis, three intersects it. I mean, what are the coordinates of this point? You said three comma zero. No, well, five from the positive y-axis. It means so much, so much, so much. If the length is five, then this point would be I figured out the coordinates are zero point five. Now we want the equation of this circle. Okay, we Let me take a look and give this a name. I named the point "Take it". I named it B. I named it C is now A R B I for our convenience. I connected this point a little bit. I connected the dots. Okay, tell me which corner is this CB 90? Degree No, at the corner. CB it is 90 degrees No and it must be a circular angle. The point is on the circumference of the circle. The circle passes through the origin, right? Since it says the circle is centered at the origin Then point C is on the circumference of the circle. So this is the CB at the top, this corner. If the angle of a circle and the angle of a circle are 90 It's a degree, so we can't say it. Do you understand what a semicircular angle means? What kind of angle is this CB angle? You would say it's a semicircle. The angle of the semicircle is now a right angle. The angle of a semicircle is formed by two points on which What size are they standing on? What does AB actually mean if it is not a vertex? What you told me is that AB is actually What is this diameter now? I'll give you two endpoints. If there is a given endpoint of a diameter If the endpoint of the diameter is given From there to there, from there to the circle What's the job of children to solve equations? You would tell me X minus X one into two. X minus X Two plus Y minus Y one into two Y minus Y equals zero. Just simplify it and it will work. X squared minus three x plus y Square minus five and zero, what are you? Notice that the value of C here is zero. It has to be because it goes through the main point. Say no. I think I understood, look at the answer. Look at x^2 + y^2 - 3x - 5y c. Value is zero - 3 is the identity of this three Is it the distance from the x-axis? As far as the positive side is concerned, these five What is the identity from the positive side of the y-axis? If you don't give me three, as much as I'm spending now Let's take any value and put it on the x-axis. This is subtracted from the so here is the three in place. This wouldn't have happened from the minus XY axis. Part B is being cut off, so it would be B here. What would happen here? It's common sense, but... Here is a nice MCQ shortcut. Can you take a shortcut? Do this, I will always remember it. A circle is a circle that is basically made up of dots. which passes through the origin and From the positive x axis and the positive x The part from the axis is positive x axis from the axis. The part is being cut, which means the length is this much. The length of this is a and positive It means that part B cuts the Y axis. This much length means this much length. It is zero comma. P then of course this is this and that two What you get by connecting the dots Is that the endpoint of the diameter? Is that What will be the equation of the endpoint of the diameter? Just apply the previous formula like this. Something will come out of x squared plus y. Square minus a bracket Give minus give another bracket = two What will be positive in the first bracket of zero? The amount that is cut off from the x-axis is positive. How much does it intersect with the x-axis? It is cutting. What will be the positive y in the second bracket? A few as far as they cut from the axis Let me give you an example. Now I told you. Positive A intersects the positive x-axis. The story is intersected by B from the negative y-axis. If you understand, then tell me what the equation will be. So just tell me what the equation would be. I'm changing it a little but This will cut the positive x-axis at A. I said but the negative x axis is negative. I'm a little color, B intersects the Y axis. I mean with the color yellow. Yellow ancerta positive x axis to A What positive x will fit in this bracket? How much is cut off from the axis? Will the positive y axis fit in this bracket? How much is being cut from the Does negative mean negative p here? The negative y-axis means that the day is If five from the negative side of the y-axis If it is cut, then it is minus five from the positive side. I can't say it's cut, of course I can. Of course I can. Now I'll give this too, negative and give this too. Negative x axis is OK. From A intersects the negative y axis to B. It's cutting, so what's our answer? Everything else will be the same as in the bracket. Things that aren't inside will change. What will be positive in the first bracket? The amount of negative x that intersects the x-axis The positive x-axis is intersecting the axis at What is being subtracted from? So B intersects the negative y-axis. From the positive y-axis to the negative b It's clear. Okay, these come up in MCQs sometimes. Can you solve this question? Let's solve the puzzle in a very simple picture form. The solution can be done in the form of a picture. This is how the circle of this square is. The circle you want is the positive x-axis. From B will intersect the positive y axis from B. It will be over then the work is done then the work is done everyone To understand If you could tell me, I understand, okay. We will now move on to another concept, which is The position of a point relative to a circle is the circumference of the circle. The position of the point relative to this is called a The circle is now a point. Where is this circle? It could be somewhere outside. Can you guess here P of X one comma Y? One can be completely outside, on the perimeter. Can be anywhere or inside How can I check? It's very easy. You're the one. The coordinates of the point are the equation of the circle. Let's sit on the left side of our circle. Notice the equation, we have Notice the left side of the equation of the circle. X squared plus Y squared plus Twice G X Plus Twice FY Plus three equals zero at this point. The coordinates are here. If you put it on the seat, it will be zero. If this is equal to this, then Two zeros means the equation of the circle is He is perfecting it on the periphery on the periphery And if you give a greater number and it becomes zero, then you are out. Do you understand that x1 y1 is the coordinate of this point? Why are you sitting here, X-One? Square Plus Y One Square Plus Twice G X One Plus Twice FY One plus C is equal to zero. Put the lace on the circumference, if it is zero, put it inside. If Greater gives zero, it's very easy outside. Formula Look at what the next one says. Show that this How is the point located on this circle? How can you show this point? The coordinate is on the left side of our equation. Sit down, sit down, what happens to one square plus? One square plus four into two One Plus Six Into One Minus 12 So what is 12 minus 12 equals two? Zero means this point is on its circumference. What is it saying then located above? What would you say then? It says that the diameter of the circle passing through this point is There is a desire on the other side of it. You understand the story of coordinates, it's a funny number. But let's say there is a diameter through this point. The diameter that has been drawn is ours. The coordinates of point A are What is the diameter drawn through this point? It means that it has gone to the other side. You are asking me for the coordinates of the point. Are you the center of this circle? Know the coordinates, know the equation of a circle, remember Knowing the equation of a circle means knowing the equation of a circle. You know everything, what is the equation of this circle? So what would you say is the coordinates of the center minus Two comma minus three So, minus two commas minus three x's. Divide the coefficient by minus two to get the coefficient of y. Divide by minus two is the midpoint of A and B. So you take it out from the center. I'll bring it, I'll leave it. Even kids can do it, even kids can do it, right? Yes, we are moving on to our next concept. The next concept is to notice a circle. A circle is a circle whose equation is such that It looks like C equals zero is correct. The left side of the equation is C and a Straight line The left side of the straight line is L. Equals to zero, you understand that A is X plus B Y plus C does not equal zero. That's A X plus B Y plus C. I expressed this as their intersection point. These are the two points of intersection between the two. The circle that can be drawn through the intersection point Tell me how many circles can be drawn. Two dots make billions of circles. Can you understand how to draw? Can you tell us the white circle and the white The straight line is the intersection of the two points you are. You get a crore with those two dots. Drawing a million circles Those billions of circles will go to their common The equation will be a circle but not a circle. Notice that the yellow circles You see two yellow circles, but this one is white. The intersection of the circle and the white straight line It takes billions of circles to draw like this. Their general equation is possible. Their general equation is possible. The equation will be C plus K into L equals You don't know the value of two zeros yet. Another condition for finding the value of K is It takes a lot of common sense to find the intersection of these two. How many two two two dots specified It is not possible to draw a circle, there are many circles. It can be drawn because two conditions are circles. The equation requires three unknowns and three conditions. Two conditions are given, one unknown quantity. It will be okay, you understand. Right. Okay, we'll need these. Later, after that Look at the circle that intersects the two circles. Our two circles are the equation of two circles. Take another one like this: C One Equals Two Zero. Assume C2 equals zero C2. Equals to zero with their intersection Tell me how many circles it is possible to draw. How many circles are drawn through their intersection points? It is possible to draw billions of circles like the previous one. It's not possible because two dots are two. Circle cannot be fixed with a point. It is possible to draw billions of such yellow circles. What will be their general equation? These yellow circles are their general equations. C1 plus K in situ = 0 K Another condition in the question of "You don't know the value" The value of K will be given from there. I can explain that it can be brought. Okay, we're going to Ank, we're going to Ank. I'm going to read the question. It says find the equation of a circle whose The center is six commas. Zero and which is this circle is another line Do you understand what the intersection point is? Coscenta You understand one. A circle is a circle with a given center. And that's another circle and a straight line. The story goes through the intersection point. Don't worry, this is one of ours. Circle This is our circle. Okay, wait, let me show you a little differently. Let me show you a little differently. Let's take a circle. And this is a straight line. Okay, this white circle is this one. The white straight line is that one, it's new now. We need an equation of a circle that is these two The intersection of these two passes through the intersection point. You give. You see, we need the equation for the yellow circle. Whichever is which is x equals two Three lines x equals three lines and X squared plus Y squared minus four X equals zero for these two. passes through the intersection point and its The center and the yellow of our yellow circle Give me the center of the circle. It would say that the center of the yellow circle is as given. That's how to make six point zero. To do in many ways Whenever you look, whose equation is it? I need the yellow circle, not its center. There is Only by knowing can the work be completed. Somehow you want this white line and this Find the two intersection points of the white circle. Can't you figure out how to solve the two equations? If you solve it, you can see the value of x. There are three given, so that means these two. Bindu's corner already knows the value of x. If you put three here, you can find out the value of crores. It means that these two points You can find out the coordinates, that means. If you find a point on the circumference And you already know the coordinates of the center. That's the distance between the two. Radius No, the center radius equation will work. This is one way, this is one way, but this is This equation would be complicated if x equals Two or three would be a little more difficult, hold on. X plus three y equals four. Line The equation can be done like this, just do it a little. For those who can't do it this way, it's done here. Okay since the center coordinates Given the yellow circle, the equation will be I know the coordinates of its center and its radius. How do I find the radius of a circle? If we know any of the above points, we can The yellow circle is this white circle and the white is simple. The white circle passes through the intersection of the lines. And the intersection of the white straight lines is their equation. Solve, you have found a point on the circumference. You have found the center radius, the work is done, another one. This number can be done in another way. Let's go for a bit. Look at me. It would be yellow. The circle is yellow. Notice the circle. It's yellow. The circle that is actually x^2+ y ^ 2 - 4x = 0 is the circle and x - 3 = 0 is the circle. It passes through the intersection of straight lines, right? Tell me our yellow circle is this simple The intersection of the line and this circle passes through So one is a circle and the other is a straight line. What is the equation of the circle passing through the point of intersection? C plus If K into L equals zero, then what? You could tell me x squared plus y. Square minus four x plus k into X minus three equals zero then What is reading? There is x squared plus y. Square x square plus y Square plus X Into K minus four x Into K minus Four minus three equals zero. That's it. But the equation of the yellow circle is an unknown. There is a zodiac sign here, what is that, who is that, who is that? The question is, how do I find out the value? The yellow circle will have one condition. I have not been given the coordinates of the center. I would say So the coordinates of the center of the yellow circle are Given the equation of the yellow circle, this is it. What is the center of the yellow circle from its equation? You told me to take the coefficient of x and subtract it from it by minus two. Divide the coefficient of x by minus two. Share Divide the coefficient of the function by minus two. This is what we are given to share. There is So, if you don't tell me, then what's left? Minus four by minus two = two six. From there, what is the value of K minus 12? Plus four or minus 8, who's done? The value has been removed, please put it here. So, is our yellow circle the equation? It will become x ^ 2 + y ^ 2 + k - 8 - 12 Then it happens + 24 There are advantages to doing it this way. The advantage is that x equals three or No matter how complex this equation is, If you do this, it will be easier than doing it before. The problem will be whether everyone understands the previous process. I hope everyone understands, if you could tell me. We got the last of Chapter 41. Topic Parametric Equations You must be familiar with You hear that X and Y are a third variable. Notice that it is expressed with X and Y is expressed with theta This is the parametric equation of a circle. So from parametric equations to normal equations How do you get the third variable, or theta? He has to be removed. How can he be removed? You told me, look, X equals X. Don't give Kos Theta in the afternoon. Y minus C equals a sine theta. Don't give it, take this number one, take this number two. Number one, number two, you. Square and add the numbers one and two. If you square the numbers and add them, What is x squared plus y squared? Equals to a square into cos square theta Oh, what did I write? X minus B is square. Plus Y Minus C Whole Square Equals Two a square Into cos square theta plus sine square Theta Cosine Square Theta Plus Sine Square How much is the value of the theater? So why don't you just sit down and do it? Minus B Hole Square plus Y minus C hole square Equals to a square, so look, this is a The equation of the circle is not coming, whose center is at what distance? What is the radius of comma C? So, given the parametric equation, what do you do? From there we can take it to the normal equation. Look at the math, is it given? Let's say the parametric equation of a circle is where What is the parametric variable theta you are looking for? The center radius of the equation is Habib Ber. Until then, we have many things to do. I don't see the math where the equation of a circle is Various information would have been given from there. How much does the Y axis intersect the X axis? How much is being deducted from If Habijabi does not give that equation directly, You give the parametric equation in the form Couldn't you have done it faster? Let's take the parameter to its ideal form. That's how you get the parametric variable theta. Theta ke hotay dyba Theta ke bhagay dyba How could Bhagavad Gita tell me that x - 1 = three cos? Theta is the trigonometric function of the The formulas are then y - 2 using those. = How many 3 sin theta add one to two squares So what remains is x minus one. Square plus y minus two holes Square equals three squared. Take this. One, hold this, two, center, one, comma, two. The radius is three, if you want it that way. Even in the ideal shape of the GFC Then how far from the x-axis can you go? How much of it intersects the y-axis? Area is how many pi and square is many things I realized it was possible to find out, so you told me. Through this our chapter 41 It End Ok we will now start Chapter 42 Chapter 42 is mainly about tangents. It is discussed that tangent tangent is a simple A line is a straight line, and many things in the chapter There is a relationship here with you, a straight line. If you read the chapter well, it's absolutely It will become a chapter for the children. So we are starting the chapter. 42 So we will now start Chapter 42. Essentially tangential The tangent to the circle is before that. We have not seen one circle and the other a straight line. What state can they be in? The circle is three positions away from another straight line. You might notice that the first one is a circle and the second one is a line. The lines are far apart, far away. There is nothing in common. In the second case, the straight line intersects the circle. Touch In this case, this straight line is called The tangent to this circle or Tangent and everyone's favorite, our simple The line intersects the circle. This is called Either this straight line intersects this circle. Or the English term is two seconds. The two are intersecting at the point, noticing It intersects at a point and when you touch it, what happens? Just do it to the point. Okay, now one circle and one straight line. How will you find out if you are in that situation? From the center of the circle to that line The length of the perpendicular drawn is the center of the circle. The length of the perpendicular drawn from the line to the is greater than the radius of the circle, then The first condition is on that line from the center. The length of the drawn perpendicular to the circle If the radius is equal to that straight line, A tangent to a circle is very important. We discussed this case at length today. I will do it, the questions will come from here, okay? The perpendicular drawn from the center to that line If the length is greater than the radius of the circle If it's small, then it's a cutter, okay? Now if the equation of a circle is given You find the center radius from there. I can't do it, so I'll draw this straight line again. If the equation is given from a point The length of a perpendicular drawn on a line Do you remember the formula to figure it out? Can you know the equations of circles and straight lines? If so, what is that circle and that straight line? You will have to take it out if it is in a state. Parba Okay, the most important of these. is the condition for the middle one to be tangent You understand that you will memorize it completely. Drawn on the tangent from the center of the circle Equal to the radius of the perpendicular circle, correct. From the center of the circle From the center of the circle Length of the perpendicular drawn on the tangent The radius of the circle is equal to the circumference of the circle. I understood it to be equal to the radius. I write it down formally. Wait, this is tangential, okay? I colored the circle a little white. Show him that you are the center of this circle. I can't do it, so the equation of the circle Finding the center radius if given If you find the center, you can also find the radius. Is the radius of the circle also an equation of a circle? Now this straight line can be derived from Take the equation: X plus B Y plus C equals two What is the condition for zero tangent to be from the center? The length of the perpendicular drawn on this line is the circumference of the circle. From the center to this, the radius is equal. Could you tell me the length of the drawn perpendicular? Minus AG remembers the straight line chapter Minus AG minus B F plus C Divided by root over of a square plus This is B Square. What is the distance from the center to this point on this line? The length of the perpendicular drawn above is now if It must be tangent to what radius? Equal The top line will be at the top again so that Goes in. You understand, even if you beat him, he won't come out, yes? Once you memorize it, you won't be able to get it out again. There are questions from here, let's take a look. Let's take a look and see what a line says. If you want the condition to touch this circle Can you draw a small picture? We don't have a problem even if we don't draw pictures. What equation would you give for the circle? X Square plus Y square equals A. Square it will come in a row or How many zeros will be given in the MCQ for the center of the circle? How many zeros are there in the center of the comma zero? What is the radius? Tell me the radius. The radius of this is So what is the equation of this line? Bring aside M x minus Y. Plus three equals zero, this line is this. What would the condition say for touching the circle? If you hold this and if you hold this Take the question. According to this line, this circle should touch this line. Condition O M Equals to Radius of Circle What will be the distance from the center to this line? The length of the drawn perpendicular is zero minus zero. Plus C divided by root over one Square plus M squared is radius. So what's left is the modulus of C. Stay on top and send it to the other side. Into root over of one square plus m Square, you can make it square on both sides if you want. Can you please remove the modulus sign? So what remains is this, is this ours? You can memorize the desired conditions very well. Comes into place In MCQ, this line touches this circle. You just have to remember the condition to do this. This thing is at that radius from the center. What am I saying above, on that line from the center? The length of the drawn perpendicular is the radius of the circle. Radius of equal circles Equal Look at what it says, it's very important. The question is, this line is an unknown on this line. There is a zodiac sign B. A condition for finding an unknown quantity This line needs to touch this circle. But there is one condition: this line is like this Asks to find the value of tangent B of the circle Story If you understand the story, then our circle What if the equation of the circle is given? Everyone tell me what the center of the circle will be. What is the center of the coefficient of x minus two? Divide the coefficient of y by minus two. So the line that is given now is If the line is tangent to this circle, then You are told to find the value of B. The equation of this yellow line is three. X plus B Y minus one equals Find out the radius of two zeros. Our white circle, our white circle The radius of the circle will be the root over of D. Square, you are a little bit in Bengali. Write the coordinates of the center of the radius. It's G squared plus F squared. Minus C 16 and one 17 If you subtract four from 17, what is the Either 13 routes over 13 singles Okay, take its coordinates and this. Find the coordinates of the point. Question. According to the question, this line intersects that circle. Does not touch means the value of O M Who will be equal? And it will be equal to this, apply it many times today. If so, how many points from the center of this line are there? The length of the perpendicular drawn above is four point one. What is the length of the perpendicular drawn from it to Well, look at the three seedlings 12 Stand up, stand up, three seedlings of 12 Y. How many seats are there in one plus B minus? One divided by root over root over Of three square plus b square root over The coefficient of X plus the coefficient of Y is Square equals to radius what root is 13 Do you need the value of B? Do you need the value of B? If you multiply the value further, Modulus of 11 minus B 11 plus B Sorry. Modulus of 11 plus B equals to root Over of nine plus B Square into Route 13 Now square on both sides Do it. So whatever is left is eleven plus b. Square equals 13 into nine plus b. Solve the square by simplifying it. Will solving it result in a quadratic equation? A quadratic equation of B will be obtained. Solve the quadratic equation B. The value will be two and minus one by six. Solve it. I figured out the rest myself. Neo A plus B is the formula for the whole square. Break it up, put it aside, or do a midterm. The last work, the last two tangents are coming, notice. You are doing the same thing, watch out for the next one, it's the same thing. Show that this line is tangent to this circle. Does the condition of being come up many times at all? Simple this line this line this circle The condition for tangent is the center of this circle. How many decimal places are there to find the radius? How much is the center of this comma zero? What is the radius from this center to this? The length of the drawn perpendicular is equal to the radius. From there Let's see, try it yourself. Important, but if not, on the next page. If you can't do it, it's on the next page. Okay, same, same, what's another number? What is the value of this line touching this circle? Will the condition of being tangent be its center? How many five point zero is the radius from the center? How many five five point zeros from this? The length of the drawn perpendicular is equal to five. Who will bring the value from whom? Two values ​​come out, 40 and minus it. But I'll do it, I'm going to the next one, the same thing. What does this line say to this circle? The condition of being touched by touching What does PQ say to show after applying? The point lies on a circle. How can we show that the point PQ is from the condition? Located on a circle, OK, done. Notice that the condition for touching him is coming. Do this. Look at this now, it's the equation of a circle. Like this is not the equation of a circle where X and Y have P and Q instead. That means we can't say PRQ. The point is actually the circle on which this is located. I understand that it is located there. We are moving on to the next type. What is the chapter? With tangent, the equation of tangent is A very important topic. An important topic, question from here. The chances of coming are higher than before. It's actually tangential, just like the type. What is a straight line tangent? What is a tangent? A straight line, you are straight in the previous chapter. You're reading the lines, so tell me, it's simple. There are many ways to find the equation of a line. That It's all here, so you can calculate it. Find the equation of a straight line from all of them. Two conditions are needed to do this: two points. Given a straight line passing through two points, Remember the equation is a point and the slope. Then if the shield and the interceptor are given or All intersection shapes can be applied here. Okay, the concepts of the equation of a straight line. It will take a while, there are basically three categories here. The first is from the condition of being tangent. What does the condition of being tangential mean from the center? The length of a perpendicular drawn on a line If is equal to the radius of the circle then Find the equation of a straight line that is tangent to it. How many conditions are needed to do two? One of the two conditions is which one Another condition is that a circle will be tangent to it. Our first question will be given in the question. The second type is the circle type. The equation of the tangent at the point above is another is the equation of the tangent from the external point I'll watch these two later, but for now, the first one. Is the type subject to the condition of being tangent? Tangent The equation is again the equation of a straight line. How many conditions are needed to find two? Conditions Okay, one condition is fixed in this chapter. Another condition is that it is tangent to a circle. If you give it, the work is finished. Read the question in this circle. The tangent drawn is parallel to this line. It says two to find the equation of the tangent. The condition is that you get a straight line or a There are two conditions to find the equation of the tangent. You got one condition: it's parallel. Another condition is that it The equation of the tangent line will need two. The condition is that I know the work is done, the work is done, okay? Five x minus 12 Five x minus 12 Equals to 9 is any line parallel to it. The equation of the line is like this Otherwise, you would write this straight line in Bengali. The equation of a parallel straight line is this It will take We are not calculating the value of what is What is the value of the line tangent to this circle? I remember doing it a while ago. Look. So look, look, look at this. This line was an unknown quantity. Find the value of tangent B of the circle. I haven't been converted to the previous type. You would say that this is the equation you need. The equation of the tangent is What is the value of K from here to here? Find out since you are the tangent to that circle. If you want, draw a picture. You can draw pictures, it's easy to explain. What is meant by drawing? Easy okay Okay, okay, just a little bit. Be the center of our white circle. How much would you say four point five or with an X? Minus the coefficient of five y by minus two. Two divided by five is correct. What is the radius? Take out our white circle. What will be the radius of our white circle? The radius will be written in Bengali. The center of the circle is this radius, this root. Over of 4 ^ 2 + 5 ^ 2 + 8 - Three would not have said. Minus-8 So many units, what happens? Calculate a little. Root over four squared plus five Square plus 8 is Now if we apply the condition of being tangent to seven It will be, hold this and hold this, this is the yellow one. The equation of the line is 5x. - 12y + k = 0 To find the value of k Who can figure out its value? If you can, then it's okay. Question by question. What? Will be I would say You are the ones who are roaring outside. I hear that Kalbaisakhi is coming. I think it's okay. According to the question, what will be the value of OM of the circle? The diameter will be equal to the radius. Now what is the diameter of the circle? The perpendicular drawn from the center to that line Length Five into Four Five into Four Minus 12 Minus 12 Into Five Plus What is the value of K R and what is its value? Seven divided by the root over five Square plus 12 squared down is 13. Below is 13 seven 13 how much would you tell me seven 13 Being 91 modulus of four five 20 minus 60 Modulus of K minus 40 K minus 40 How much is equal to? 91 Then if the modulus sign is negative, then 40 equals two plus minus 91 where Which comes from equals to 40 plus minus 91 So if you add 91 to 40 Our K value is either 131 or minus How many times would you say that if you subtract 40 from 91, you get 51? Two values ​​of minus 51K If two values ​​of K are coming, then We have two equations for the tangent. But X five x minus 12 y plus 131 Equals to zero is a five x. Minus 12 and minus 510 Why did two come? You said the answer is very easy, pay attention. Attention Pay attention, pay attention, I'll show you a little. One is parallel to a straight line and the other is The tangent to the circle is the straight line below. Take the yellow straight line, five x minus. 12 Y Equals to 9 Eight Parallels And the two tangents to this line can be found No, tell me, I can get this one, but I can't find another one. Can be found above. So what will be the perpendicular distance between them? 14 is equal to the diameter, so the type is understood. The equation of the tangent line must be found. A straight line tangent to the condition is a straight line. Two conditions for the equation of a line are two. If there is a dot, there are two conditions, there is one. The condition is fixed in this chapter. Which one is he? The tangent to the circle is, so if we give another condition, Is there another condition here? You said this line is parallel to this line now. You could also say this without giving a parallel. Perpendicular to the line, so two more tangents I would have gotten it or not, any other way. I could give you any one of them on one condition. So Let's see what you guys are doing in the straight line chapter. You've read all of them, just one condition. Just add it here and it will say "His shield". This many degrees of angle with the center or x-axis Creates or from the X axis and Y Say something by cutting equal parts from the axis. Just give it and then take care of it. Make the tangent of this circle on this line. You see, there are two conditions given here. If it is perpendicular to the line, tell me. What would the equation be like? Four x plus three. Y plus K equals zero K. Don't know the value, it's on that circle. Who will find the value of tangent from there? He has been made here, who is here too? Why don't you draw a picture to get two values? I understand, I understand, I understand. I found that the next one is a tangent to this circle. Condition positive Y is 30 towards the X axis. Are you paying attention by making a degree angle? Are you paying attention? Then it's tangential. What would be the equation Y equals to me? Tell me, tell me, what is the slope of the tangent? What is the slope of the tangent to the line 30 or 30? What is the slope of the tangent to one by root three? Then draw thirty or one by root three. What is the equation of the tangent line? Y equals to M. Multiply the root by x plus c root three. Three Y equals two X plus root 3C take it aside. X minus root three Y plus root three C equals zero, what is the value of C? How do you find out the value of this or that? Apply the tangent condition of the circle to C. The value will come out, the rest is homework. The rest Homework, look at the next number. It says "Wait a minute, there's something missing in a circle." Here square square is tangent to this circle. Two axes are equal in sign. Section intersection You've made two conditions, but this is what you give. What does it mean since it comes from the axis? This means that the intercept here is from the x-axis. What is the equivalent of the distance from the y-axis? If it's not the same sign, tell me. X plus Y X plus Y minus A = to zero, this is the equation of the tangent to A. How does the value need to be tangential? To understand the condition of being tangent to I got it. Okay, now we're on the next type of tangent. We're going to look at the second type now. is the tangent to the circle at a point on the circle Equation means a point on the circle. The tangent at that point will be given How do we find the equation then? Is from some external point Equation of tangent You can ask questions, brother. Tangent from any point in the interior is What will the equation be like? Say, at a point above, the circle Tell me how many tangents can be drawn. One, we will now find this equation. The type is this and the other one. Two tangents to a circle from a point It is possible to draw from an external point. Two tangents to a circle Drawing I understand it's possible, wait a minute. I'm drawing this and this is ours. The last one is now somewhere inside. Draw a line from a point but not a tangent. It's not possible from here, no matter what you draw. That will be a deal breaker for us now. Type I'm looking at the question. Read this circle 4-1. Equation of tangent and perpendicular at a point Just reading the question, it becomes clear that If this point of ours is not on the circle, But this is not how this circle would be said. You can still check this at the point. How do you know if the point is on this circle? Check and confirm its coordinates if What if the left side is equal to the right side? The above point is a bit faster than before. The picture is not drawn, let's take this. How much is the center of this circle? I would say this circle and the center would be 3/2. The center of this circle will be this circle. The center will be 3/2 comma. -5 So we didn't find the radius. At which point on this circle are points 4 - 11? 4 - 11 4 - Equation of tangent and perpendicular at 11 points There are two things: tangent and tangent. The equation of these two is perpendicular and normal. So, what are the two tangents and perpendiculars that we have discussed before? Let's try to understand what tangent is. Everyone understands, right? That's why I'm so excited. Perpendicular means at the point of contact. Let's say the perpendicular drawn on the tangent is Perpendicular means at the point of contact. The perpendicular drawn on the tangent is this If the point of contact is on the tangent here If you draw a line, it will look like this: the sky. The color that is called the vertical or There is a normal theorem that at the point of tangency If you draw a perpendicular to the tangent, it is the center. It will pass, it will pass through the center, right? There is this white line and the sky-blue line. What equation is the line asking for from these two? No new formula other than using the same type Without the concept, we only came up with these two. I would say I can do it, I would say I can't do it, I would say I can't do it. I named it and I named it MR. Hold the tangent and tell me if it's okay. The two endpoints of a perpendicular You don't know, with two perpendicular points This goes to this and this is the line joining two points. Applying the equation is the only way to finish the job or You can find the slope of the perpendicular if you want. We assume its slope is perpendicular to M. The hearts of millions would tell me how much the shield would be. The heart of the two arms is not so minus. Five plus eleven crores of people are in love The distance between the two sides is three times two minus four. Calculate it a little and see what will remain. That is minus five plus eleven. 15 minus four minus five minus 12 by Five, whose shield is this? You would say it's vertical. If the slope perpendicular passes through this point, then What is the equation of the perpendicular to a point? Apply the equation of the line y minus y. One equals two equals two. Into M Into X Minus X One Now If we simplify this, what do we The equation of the perpendicular will be found now. If you tell me the equation of the tangent, what will happen to me? Everyone, tell me what the equation of the tangent line is. Is the tangent AB or the tangent and perpendicular? The slopes of the tangents are perpendicular to each other. Will the slope product of the perpendicular be minus one? If you reverse one negative, you get another. If so, I call it the slope of the tangent. Let's think about the slope of the tangent with primes. It will be negative by making the slope of our perpendicular They have to be inverted because their product is minus how much? One that happens 12/5 5/12 Again our tangent is four commas. Minus eleven is going through the point. What would be the equation of the tangent to the curve Y? Minus Y One Y Minus Y One Equals Two equals two M into X minus X realized this was a new one. Now above a circle without a formula The equation of the tangent at a point is We have a nice formula for that. Let's take a look at the formula. Let's take a look at the formula. Let's see, this is our equation of a circle. The standard equation is in standard form. The standard equation of a circle is now To remember a point on this circle But what is the point? The tangent drawn at this point above What would be the equation for some replacements? What replacement do we need to do? X square instead of square X one Y is going to be installed in place. Square is going to be replaced by Y Y. One X One Y One is what The tangent to a point is the point above it. It would have to be, but its coordinate is X. I will put it in place. I will put it in place of X. X plus X one by two Y's I will put Y plus Y in place. To make these replacements one by two What remains after that is that. The equation of the tangent drawn at the point is then What would be the equation of our tangent? X One Plus Y Y One Plus Twice G Into X Plus X One Buy Two plus twice F into Y plus Y one by two Plus C equals two Zero, everyone understood, please tell me a little bit. It has gone up, it is time to bring it down. I am making arrangements. Let it be after all these replacements are done. That will be the X on the circle. One comma and one dot. Let's look at the equation of the tangent. We will do the math again. Okay, let's do this math again. You want to say that the tangent at the point above is Just a quick reminder for now. Let's find out the tangent, just the tangent for now. What is the coordinate of our point in the equation? Four point minus eleven, please check. Take that four point minus eleven. What is the replacement for the above point? X should be replaced with X one. Is X one X one instead of X? What is Y Y instead of Y square? One is not X squared, instead of X One Y Square instead of Y Y One just replaces X with X. What will happen? Plus x one x plus x one by What will happen to me in place of just Y? I would say Y plus Y one by two Y. Minus 11 divided by two equals 15. But there will be no change in what is coming. If you simplify it, what you get is Four comma minus eleven is written in dots. Simplify the rest of the equation of the tangent. Do it then, the equation is perpendicular. It can be found perpendicular to this and this This formula is correct, it passes through the point. I understand that you need to keep that in mind. Let me write some replacements nicely. It will be done in place of X square. In place of X one Y squared Replacing the Wii U One X Should we replace X with X plus? What is the place of X one by two Y? Basaba Y is replaced by Basaba Y. Plus Y instead of one by two Y. Basaba Y plus Y one by two R A constant term is just a constant term. So constant It's exactly like the post, see what the next one says. A circle is a circle. X of this circle is equal to At point two, the x of this circle equals two. Tangent at two points The number is only given in the square root, not in the crores. If you need x equals two, then I need it. They said x equals two if two is two squared. Plus y squared equals 20. Equals to 20 from there y squared The value 16 is not y squared equals two. What is the value of 16 Y plus minus four? That means the equation of the tangent at two points is You understand the story, they're asking you to find out. The dot is two comma four. The equation of the tangent drawn at a point is another It is two comma minus. Where is the other four? The other is two commas. What is the formula for minus four? I learned that, apply it to X square. X in place Instead of that, Y Y One and constant. Set aside x as a constant. X one Y instead of square Y Y One R instead of Square Constants aside Come on, I understand, at one point above. Learning the equation of the tangent line Okay, the next type is the outer point. From a circle to a circle outside it Two tangents from a point Our current target can be drawn as follows: What is the equation of two tangents? Everyone understands that our current target is to find out. From what external point are the two We can draw tangents to their equations. Out I will do the question. Read this from minus five point four. Do you understand the language of the question on the circle? From point to point, from this point, it means understanding. This point is going to be an external point even then. You can check it out while you're sitting here. It's just a matter of whether it's greater than zero. Let's see what happens when we put an external point. Greater gives zero, which means external. How do you find the equation of a tangent to a point? Let's see what we can do, so take this. The circle we want is this circle. How far from here is the center of this circle? The center of this circle is one comma two. The center is one comma. The center is one. Could you tell me what the radius of the comma should be? The radius of the circle will be the root over of G. Square plus F A square minus C then How much is two units? Two units, okay, now this is an external point. Take this external point as an external point. What is the point minus five point four? Minus five point four is the outer point. From this From this outer point on the circle to this circle This is one of the two tangents that can be drawn. Tangent And this is a tangent, the equation of these two. Me Two conditions are required: tangent is a straight line or not. What are the two conditions required? Have you got it? But this is the first time this point has passed. Conditions No and the tangent to that circle is the second condition We assume that this point passes through The slope is M. We assume the slope It's M and minus five point four. If it is a point, what would the equation be? Y minus Y one equals M. Into x minus x then on one side if Send M X minus Y plus five M. Plus four = 0 so no me You said plus five M plus four = 0, that's it. Desired tangent The equation M will have two values, assuming Two tangents are going, so what is my target? Now how do I find the value of M? From the condition of being tangent to being tangent What is the condition for the tangent to be from the center? drawn on any tangent from The length of the perpendicular is equal to its radius. If you think about it, if you think about it, Question Opinion question According to O M equals to R from the center What will be the length of the perpendicular drawn on it? The center is said to be one comma two X. Instead of one Y, put two. Modulus of M minus two plus five With M plus four modulus I gave it to you, I understood that this much is divided by Divided by root over one plus m Square equals to our radius. What is the radius? The radius is two. This will solve the value of M from here. Look what happens here, the modulus of Six M plus two equals two into two roots. Over of one plus m squared goes to two commons Divide it by two, then what remains is three. M plus one modulus of three M Equals to root over of one plus m Square both sides of a square, what is left is 9 m? Square plus six m plus one equals Two One Plus M Square One One Ghachang One M square is subtracted from nine M squares. What if I give 8 M? Square plus six m equals zero. What is common, tell me twice if it is common. Now four M plus three equals zero. From there, two values ​​of M come out. Equals two zero and minus three times four. M equals two zero and minus three. Bye bye everyone, I understand that this is M. There are two values, these two values ​​are coming. You put two tangents in the equation. The equation works. Come on, tell me, tell me, can't you all do the rest? You said you can definitely do the rest here, M. If you want, just set two values. There is a value of M that will be zero once. It will be minus three bye. Four does the rest. Okay, important, but the same. Another one The number is drawn from the origin to this circle. The equation of the tangent drawn is the origin. What will be the equation of the tangent if we go through Let's say y = mx. No, take it aside, M X minus. How do you find the value of Y = 0 M? Draw a tangent to this circle at its center. The length of the perpendicular drawn on it from From there M is equal to the radius of the circle. It will have two values. The rest is done here. Now we are dealing with the equation of a circle. Some terminology, some short terms, we Let's see, this is the standard form of a circle. Nor is the standard equation of the circle standard. From standard to S this standard equation The left side is the left side of this ideal equation. The side is expressed as S. What does the ideal equation of a circle mean? S equals What is the ideal equation of any circle with two zeros? If you want S equals zero like this Can you write S is standard? I understood that everyone was saying that it was coming from Okay, now this is the standard form. If you mark a point X on the left side One comma and one, please be careful. The left side of this standard form is you. Are you replacing X with X-One? Y one instead of Y means X, Y in standard form If you sit in place X, then left. The thing on the side is called S. What does the abbreviation S1 mean? Okay, well, and one above the circle. Point X one comma Y one there If a tangent is drawn to a circle, Point X one comma Y one there Even if we draw tangents, we don't see them. Making some substitutions in the equation of the tangent After making the replacements that are needed There is this called T tangent tangent Coming from these three things, remember a little. Rakhba SS 1 standard form The thing on the left side is called S. If you are on the left side of the standard form Put X one Y one then which one It is available on S1 and above. The equation of the tangent at a point is That's it, I hope you understand. Everyone Next, just remember the formula, this one. From a point outside the circle From a point you are on this circle Can you draw two tangents and their lengths? It will be equal, right? From an external point Two tangents from that point From that point to the point of contact That's the distance to the point of contact. The tangent length is correct. What will be the length of the tangent to the root over the line? S1 Fast converts the equation to ideal form You have to bring the equation of the circle, you understand, fast. Then it must be brought into ideal shape. Then the left side over there is X. In place of X one Y one Y If you install one, you root it, that means X one comma Y one X one comma The tangent drawn from the point Y1 is The length is both the route over the S1. Equal root over of S one, do you understand the point? If it is above, then it is zero tangent. If the length is zero, then the tangent is drawn. It won't work, it will come out as a complex number, right? This value is not zero if the point is above. Think about it for a moment. So our first question from this point The length of the tangent drawn to this circle is yours. The first thing to do is to put the equation in standard form. You wouldn't have told me to take it, would you? Off S One Standard Form x1 One And y1 one will be placed in place of XY. What if we take it in standard form? What should I do? Divide by two. What do you have to do if you divide by two? X squared plus Y squared minus X by Two plus three times two plus half. = Two zeros is the standard. What form is required? Now one comma minus one one comma minus Tangent drawn from point one If I take the length of the tangent, Let's express with L what will be L, will be the root. Over this standard form Here you are X one instead of X. Y one instead of Y in the seat. Basaba minus one by two minus three Y by two three Y no So plus three into minus one by Two plus half, then look how much. Look how much more one and one can make than two. Two minus zero is the root over half. Route over half the length since you You have to write the unit length since you have to You have to write, you have to write, you have to understand. Tell me if you can. So now this concept is being used in many ways again. Maybe so many centers from the origin Length of tangent drawn to the circumscribed circle The equation of so many circles is asking for the center to be given. I don't have it. I would say GF. Given the value of C, the root is What is the tangent drawn from a point to a tangent? Is the length of the route over the sea or not? You are a little Just imagine, we assume. The equation of the circle is Since this is centered on GF Value I already know the value of C. What would you say is the value of G when the work is done? What is the value of minus one F minus two C? Its value is now from the original point. What is the length of the tangent drawn to the original? From the point it means the root over of zero. Zero zero zero everything zero plus zero Plus zero plus C is how much is it? Two. Square both sides. What is the value of C? The value of GFC is coming to us. Circle The equation also works. It's coming, clear, okay, we'll do it now. Moving on to the next topic. This is happening. This is happening. Touch or cord of contact touch Or what does this mean, cord of contact? This circle is a circle. Now this circle An external point is outside this circle. Think about this point for a moment. The coordinates are X one comma Y one. The coordinates of this point are x one. Comma Y One This is the outer point. From the outer point to this circle Two tangents can be drawn, right? A tangent again this one this one I understand the touch now, these two. The tangent touches at two points. No, connect these two dots, these two. The chord drawn by connecting the dots It's a chord of this circle, isn't it? This is a chord of this circle. So this is AB AB this is one of this circle AB is also called the cord of contact or What is the equation of the chord that touches? I am not showing evidence of the touch of the chord. The equation will be T equals zero this. Equation T equals zero, which means this point If the point above were at that point, The equation of the tangent is also external if that Applying the tangent equation, which Is it possible to find out whether it is touch or cord off? Contactor Equation Clear So keep this in mind as the top point. So I'll apply the tangent equation. What is the touch or cord of contact? Finding the equation Goes from the origin to this circle Find the equation of the tangent drawn. Coming to the tangent drawn from the external point Do the equation first part yourself. The second part says touch or cord off. Contact is the cord of contact or touch. But the first part is not working. I was going to do it, but I already turned off the cord a little while ago. Contact or touch is the touch of the chord. What would be the equation? Our point was But zero comma zero is a touch of the chord. The equation will be T equals zero then X. What should be replaced with X? X and Y have to be replaced. The Wii One X needs to be replaced. It has to be replaced by X. You have to put x plus x one by two. There's nothing else, plus 20, so what's next? If you could tell me minus five x plus 20 equals zero, so what is the value of x? Four, is this the touch or the cord off? The contact equation is just a formula. Just memorize the formula. Just keep it. Okay, we're next. Dash equation of any chord of a circle Its midpoint will be given from there. We need to find the equation of the chord. Suppose you have been reading a theorem like this in class. A circle is okay, read the question. The midpoint of a chord of a circle is given. That guy is asking for the equation. Do you understand the story? The center of the circle is zero comma zero. Take any chord of the circle, any chord. Something like this will happen in any of these circles. What is this J. giving me? The midpoint is given by this J. The midpoint is two comma minus three. How to solve the equation of two commas minus three chords You are reading a theorem in a small class. Do you remember the center and the chord? What is obtained by connecting the midpoints It is perpendicular to that chord. Read Come, think about it, which center is it? If you connect the midpoints of a chord What is found is above that chord. I could explain the perpendicular to that chord. I'll give the jet a name, A B. It's Hold on, hold on to this, what do I need from AB? The equation is AB but with point M. If a point is going to know the slope of AB, It is not possible to be perpendicular to AB and M, even if you want. But you can take out the shield of O M. What is the slope of O M or M and M? The heart of the two crores is the heart of the two brothers. So zero minus minus three. The heart of the two crores is the heart of the two brothers. Zero minus two minus three by Whose shield has been revealed? What is the shield of the crores? Is this what it takes to say that its shield is okay? The slope of the triangle would tell us what the slope of AB is. Since the perpendicular is negative one by negative Reverse it and make it negative. M which is two by What is three that becomes two? Bye then to us. If you know even a single point of it, you also know its slope. What will be the equation of AB or AB Z equation? y minus y one y minus y One equals two mm into x minus X one into X minus X Okay, now let's simplify it a little more. It works if you do it. There's another formula for doing this. Yes, you understand that formula is t equals To S One, if you want, apply the formula directly. You can do this too. Acceptable, do you know what S1 is? You know that. If not, do the second process here. I just gave you the formula, just look at it. If applied, then t = s1 is this or that Take the point x1 y1 and find the midpoint. What would you do if x1 and y1 were given? Okay, now let's talk about the common chord. Common chords or common chords are generally understood. Can you draw two circles? There will be a chord like this, look at the two circles. Which is something that is common to both of these circles. A chord is a chord that has these two The common chord of a circle is called Common chord Common chord Our first circle Let's take its standard equation. What does equals to zero ideal shape mean? What is the square of x squared and the square of the coefficient? The standard equation of the first circle is s = 0 Second circle standard Equation sme = 0 okay the coefficient of x^2 y^2 is one there too One then this is the common chord this is the common chord The equation of this would be this is a straight line s - s = 0 I understand that its equation is s - S Prime equals two Zero is our first. The general formula for these two circles is To apply, the two equations must be fast. Bringing it to ideal shape and subtracting one from the other It's very easy to find out the common chords. You are the ideal of the equation of the circle. Here, taking one shape and subtracting the other Two are in perfect shape, right? You just have to subtract one from the other. Can you subtract the top from the bottom? The top is one less than the bottom. You understand, if you subtract the top from the bottom, X Square X Square Y Square Y square eight x to six x If you subtract two Xs, there will be eight. Subtract six x from x to get two x. There will be two Y's subtracted from Y's. Minus Y is 10 minus six. Four times X minus Y times X Minus Y plus four equals zero. Right OK Okay, then what does the next one say? An important number. Two Notice the circle. This is a circle. This is a circle. The circle is their common circle. Find the equation of the diameter and understand the question. We have been given two circles. The equations of two circles are also given, okay? The equations of these two circles are also given. That's just their normal behavior. General So you told me this is the common chord? The diameter of any of these simple circles will be So, tell me about a circle. The diameter will be like this, which is blue in color. This is asking for the equation of the diameter of the circle. The equation of the blue circle, everyone. Cleared. The equation of two circles is a question for you. There are two white circles given for their common The chord is the diameter of the circle they have in common. His equations are simple and fast. Fellow, we have something in common. What is the equation of the common chord of these two chords? The circle is a simple shape that they have in their ideal shape. No, subtract one from the other, two numbers. If you subtract one number from , what is left? X square Y square Ghachang two X then cut three Y from three Y Plus one equals zero so the general The simple chord equation we are not getting They would tell me that this yellow line is this yellow. How many times have I found the equation of this line? X plus one equals zero. Okay, now this yellow line is any one of them. The two points that intersect with the white circle Do they have the ends of the diameter? No, tell me, this line is different from any other. Find two intersection points of a circle. If you can throw it away, you're done. Know the endpoint of the circle from there. It is possible to find out the equation. This is a Oh, this is a way. Do you understand this equation and this equation? Solve it, solve it, then these two We can find the coordinates of these two points. What is the diameter of the blue circle? From the endpoint to the endpoint of the diameter This is one way to calculate the equation of a circle. Another way is happening, another way is happening. Our sky blue color The circle is a white circle and the yellow line is Are you the one who passes through the intersection of the lines? Do you understand the concept of our sky blue color? The circle is that one, the circle is yellow, and the circle is yellow. The intersection of the circle and the white straight line If the yellow circle comes, it will pass. Let's take equals to zero and the white line if If L equals zero, then you would tell me. The blue circle is blue. The circle is a yellow circle. And conversely, the white circle and the yellow If the intersection of the lines passes through the sky, then What is the equation of the color circle? Plus K into L will not equal zero. My target is to find out its value. The first circle is this, you could have held that. Okay, this is S equals two. Zero, you can hold that if you want. Tell me what is X squared plus? Y squared plus twice x plus Three Y plus one plus K into Twice X plus one equals zero. Now my target is K's value. Let me simplify it a bit to find out x. Square plus Y Square plus plus Twice K plus two into X plus three times Y plus K plus one equals zero. Is this our blue circle? Please tell me the equation. How to find out the value of K? Another condition is the sky blue color. What is the diameter of the circle? Two X Plus One Normal Jatta Sky Blue The diameter of the color circle is the common chord. If it is diameter, it means it is blue in color. The center of the circle lies on this line. Since the blue circle is the center of this circle, This is it, then its center must be this. Located on the line means its center This line will complete its coordinates. Solve the equation at its center. The coordinates will solve the equation of this line. What is the center of this? Tell me its center. Divide the coefficient of x by minus two. Minus K minus one is the coefficient of Y. Divide by minus two Do this, the coordinates of the center are this. Twice X Plus will solve the equation One equals zero because it has a diameter This is what happens, so who sets the value of x? Set its value to the center angle minus K. Minus one plus one equals zero. So what is minus two minus two? Plus one equals zero, I didn't make a mistake. So minus two minus who? Okay, so who's who? What is the value minus half of K? Put the value here, the work is done. Put the value here to understand that the work is finished. If we can, our sky-blue color The equation of the circle is x squared plus Y squared plus plus plus notice. So if the value of K is minus half, then set it to Into two into minus point five plus Two One Plus X Plus Three Y plus half equals two. Zero, okay, we're pretty much done now. Just a few simple ones. Concept is our next concept. The mutual position of two circles is two. The circle can be in a total of five positions. This is a circle This is a circle Two circles What can be in total five positions? Let's take a look at the position first. The location is two circles away. They will not touch or be touched. Don't do anything, this is far away. What are their conditions for staying in the state? Note the distance between the two centers. The distance between the centers of the two circles Greater than the sum of the radii is the common denominator. Sense is not the C between the two centers. One Sea One Sea One Situ Watch this The sum of the radii is two times greater than the product. If the equation of a circle is given, then Know the center, know the radius. You can't tell me on these terms. They are staying far away now. Slowly getting closer, closer Someone is slowly touching you. Are you observing from the outside what the second case is? What is the external touch second case? External contact is not the case in this case. The distance between the two centers is Load equal to the sum of the radii It is understood that the two centers are going The sum of the intermediate distances and radii They are getting closer now, which They will intersect at two points. If they intersect at two points, then their The distance between the centers must be The sum of the radii will be smaller than this. I understand that it will be smaller than Okay. The distance between their centers The previous one will be smaller than the sum of the radii. I don't understand, they'll understand later. If it starts to get closer, then a They won't be able to get inside the other one. You will notice that it will touch you deeply. But it's touching here too. It touches us deeply in this case. Note the distance between the two centers. So, the distance between the two centers is How much or not is their radius. From the radius of the largest one equal to the heart fruit Notice if you omit the radius of the smaller one. Notice that this is the big one. The radius is the radius of the smaller one from there. If we exclude it, what is the difference between the two centers? Getting distance This case is going. Okay, if they go further in, One has completely penetrated the other. In that case, between the two centers Distance Radius Interval Interval It will definitely be smaller than this time. You understand that this case is based on two points. If you don't give me enough to touch this If you don't give the condition, then the next conditions will be Get into it. Will go Do you understand that the conditions for staying here are the two centers? The sum of the intermediate distances and radii is smaller than but greater than the subtraction so that These two cases can be separated into two The circle can have a total of five positions. You are given the equations of two circles. If you are the center coordinates and Can you find the radius and the center? With these five radius You tell me what state the situation is in. How can I give the first state away? If the distance between the two centers is Larger than the sum of the radii outwardly Between the two centers touching The distance is equal to the sum of the radii times two. Intersecting at the point but going deeper No, but between the two centers Distance is less than the sum of the radii But if it is greater than the minus, then it is its Case number four will enter. Touching the center internally The distance is equal to the product of the radius of the larger Subtract the radius of the smaller one from the radius. One completely penetrated the other. There are two important cases in between. Why the number and the number four? Important here two circles touch There is a condition for two circles to touch. Externally Internally Externally Internally Okay, let's look at some numbers. This is a circle, this is a circle. The circles touch each other externally. On condition that you tell me what its center is. Zero point zero, what is the radius of this? Three. The center would say minus three comma minus. What is the radius of the root of the four radius? Over of G squared plus F squared It's 25 minus C. Are you paying attention? Are you paying attention? Do you notice that they are touching each other externally? So the distance between the two centers is That this and this c1 one situ = to r1 + r Two Seas One Seas How Many Routes Over Three? What is square plus four square plus r1 one? r1 is three r2 is two root over of 25 - c then tell me how much it is, it is It is five. = 3 + root over of 25 - c Something like that One So, so What is the value of 25 minus the C root over 25? Minus C is two. I wouldn't understand, of course, if it were two. So the value of C is 21 which is 25 minus C. Equals to four C Equals to 21 C Equals to 21 Right okay. Okay, next issue. Let's find the equations of two such circles. Make the coordinate of the center of the question so many. Read the question, read the two circles like this. Determine the equation and the coordinates of the center. It is given as long as the radius is The radius is already there and this circle Touching internally or externally Didn't you say, pay attention, pay attention? X squared plus Y squared = 9 What is its center? What is its radius? Its radius is three. This circle will touch but It doesn't say externally or internally. You understand, our second circle is ours. The second circle has its center given. What is its center and radius three point four? You know. No, you know the radius. No, the white one and the yellow one. The two touch each other, now touch. Can be done in two ways. My target is to find the equation of yellow. That means yellow is only possible if you know its radius. You know the center, now the white circle is yellow. The circle means two circles if they are inside each other. If it touches you internally and externally If it touches externally, it touches externally. If you touch it, you should say okay, come here first. From C1 C2, I take out C1. How much is the C2? The distance between these two is five. If you touch it, C1 C2 equals Two and one plus two Five equals two Five equals three Plus Three plus two from there we have I have found the radius of the second circle. So an equation for the second circle is It didn't happen like that, X minus three. Square plus y minus four-hole square Equals two squared in a more inward way If I touch you now, if I feel it deeply Touch Tell me. If it touches you deeply The distance between the two centers The reciprocal of the radius means the larger one. Subtract the radius of the smaller one from the radius. This is written as the product of the radius of the larger one. Tell me the radius of the small one. What is the distance between the two centers? The distance between the five centers Five is now the larger radius, if three is one Know the radius three, know another radius. No three if the radius is large than something If you leave it, you won't be five in life, so... One must have a large radius, not a large one. The radius of the two and one is smaller. What is the value of radius or R1? So the radius is 8, which means we have The equation will be x minus three holes. Square plus Y minus four holes Square Equals 8 Square I understand everyone. It would say that common tangents are drawn to both circles. Possible Tangent, you understand, means the intersection of two circles. How many common tangents will there be depends on On the relative position of the two circles If the two circles are far apart, then Their four common denominators are attention. You're doing it here, but there are four common tangents. There are four such tangents, there are four straight lines. Those who are tangent to this circle are tangent to this circle. Tangent shown if they Tangent in that case, touching externally If the three intersect, then the general If two tangents touch each other internally In that case, the common tangent is one and if If one goes inside the other, Common tangent zero comes in MCQ Maybe this. He would say something like that. Two circles touch each other externally. How many common tangents have you seen? Just visualize the image and you will have three Our last topic Circles in coordinates are very easy for you. You don't read Cartesian and polar transformations. So you're reading the Cartesian-polar transformation? That's it. Do you know the concept that x equals r? Kos Theta will go from Cartesian to Polar Y Equals to R sign Theta X Square plus Y square equals R. Square Just remember this, it will be this. Is a circle not an ideal shape in Karshia? You would say it is the ideal of a circle in Cartesian geometry. Now if you take this shape to polar X squared plus Y squared Instead, I'll write R square plus twice G. And what? Theta Plus Twice F R Sine Theta Plus C Equals Two Zero is a polar shape. The ideal shape of a circle is a polar shape. What is the standard form condition of a circle? The coefficient of the square must be one and There must be a square and the coefficient of the square is It has to be one, and what does square mean? X squared plus Y squared is not ideal. What was the condition for the shape to be x squared? The coefficient of y squared must be one. What are the conditions for being ideal in polar form? It must be a square and its coefficient is one. It has to be from here to the center. What will be the radius? The value of GF is the same as before. So don't change minus G minus FR. Divide the coefficient of cos theta by minus two and What does cos theta mean? X in your mind If it's difficult to think like that, And cos theta means the coefficient of x minus two. What does division by and sine theta mean? Divide the coefficient of Y by minus two. So did you get the central government's Bhuj crore? What will be the radius of the root of g squared? Plus F A squared minus C that's it. So much Simple is the ideal of a circle in polar form. The equation will have R squared and what are its coefficients? What will be the center of what needs to be one? Cos theta means x minus its coefficient. Dividing by two and sine theta mean the same thing. Y is a number that can be divided by two, including this. Let's look at the polar form of this polar equation. How do you get x squared plus y squared? Instead of R square plus R sign With Theta Options If you say it doesn't match, share it with others. So what's left is plus sine theta? Equals to zero and this is ideal. No shape, no square, no ideal shape You understand that you have to multiply by adding to take. Look at the next one, it's in perfect shape. Yes, it is in the ideal shape of the center. What will be the coordinate minus the coefficient of x? In polar form, x divided by two means R. What does Y mean in cos theta polar form? Sine theta, if you want, call them XY. You can convert the coefficient of x to Divide the coefficient of y by minus two. If you divide by two, what remains is one comma. Minus one comma minus two If you have problems with polar shapes, you just Convert to Cartesian shape. I understand right, I'm going to the next one. Next time I'm going to see it coming to Dhaka board in 23. What an ideal What should be the shape or the ideal shape? It needs to be square and multiply by both sides. Do it and square minus twice and then add up. Theta equals zero is now the ideal shape. This time, the ideal shape is coming, isn't it? Tell me what is the coefficient of x at the center of the coordinate system. What does x divided by k minus two mean? And cos theta divided by minus two. How much is this? Tell me. No, the coefficient of Y is not the center. What is the radius of this coordinate over the route? Of G squared plus F A squared minus C That's not the center, the comma zero x axis. The amount of the intercept from the x-axis How much will the two routes intersect over the two routes? Over of G squared minus C is not X. The amount of the intercept from the axis to the root What is the value of over G squared minus C? A Two root over a square minus zero Two at one I understand three, but if you give me the polar shape Don't panic, just convert to Cartesian. Do it and take the shape, take the ideal shape. There will need to be more squares to take and Its coefficient should be one. It's the same. Its center is in the ideal shape. I don't understand the coordinates without seeing anything. And if there is no such thing as a crore, how many crores would you say? How much is one C here minus three on one side? If you bring it, it will be the root over of G squared. Plus F A squared plus C √4 by two What is the number of our Ansar through this? The circle is over, we don't need anything else. Everything that is expected to come in CQ MCQ We've covered everything, so now just... Please revise the class slides once. I will definitely crack the exam, God willing. Copacabana Okay, expect more than 95% marks. I mean, if you do the classes properly, God willing. It can be easily taken to the next class. See you soon, Allah Hafez.