Welcome to another mind-blowing session on circles. Right from the definition of circle to the orthogonality. We will open up one point in front of you. Tomorrow you have to see the parabola. The line touches that parabola, cuts it.
and if it passes through the outside, then you solve the line and parabola. Take out D and see. The tangent of the M slope on this circle is y equal to mx plus minus a root 1 plus m square.
That means how much tangent will be of the same slope? The line which touches the given set of curves is called the common tangent. Quickly bring your notebook and pen and get ready to make short notes.
Hello dear children, as always from me, Jai Shri Krishna, Radhe. Radhe! Welcome to another mind-blowing session on circles. Today we are going to do the complete theory mapping of circles in this mind map session. That is, right from the definition of circle to the orthogonality.
We will keep one point open in front of you. That is, if you have watched this lecture completely, if you have been with us from start to end, then at the end of the lecture, you will find that the circle will be theoretically revised. If it is revised, then what is the delay? Get your notebook and pen and get ready to make short notes.
Let's go on today's beautiful journey on circles'mind mapping. First of all, let's see the definition and equation of circle. You all know circle. It is defined as the locus of a point which moves in a plane such that Its distance from a fixed point in the plane always remains constant. Means, if it remains constant, then the locus of that point, the path of that point, is called a circle.
Locus of a point which moves is called a circle. So what do we call it? Circle.
The fixed point, so remember that fixed point, everyone understands? Yes, you are right. What do we call it? The fixed point is called center. Fixed point is called center of circle.
Center of circle. And the fixed distance or constant distance is the fixed point. Constant distance. That is called as radius.
And circle is round. Many children also have a round circle. So from here, don't make it round. Read it properly.
So here, this is center. And this distance, this is radius. This is radius.
And this is center. This is center. Now if we talk about the equation of circle, then the equation of circle is, suppose the center is 0, 0 and this point is P, its coordinates are x, y.
So the equation of circle will be x minus 0 square plus y minus 0 square is equal to r square, where the radius is r. So the radius is r, so the equation of circle is x square plus y square. is equal to R square. Its center will be 0, 0 and the radius will be R. And if you want to take H, K instead of center 0, 0, then the circle equation will be X minus H square plus Y minus K square is equal to R square.
So that is the equation of circle. But in this case, the center will be H, K. center will be h, k and the radius will be equal to r.
So radius will be r in this case. So radius will be r. So this is the equation of circle centered at 0, 0 and this is the equation of circle centered at h, k. Now it is time for general equation of circle. General equation of circle looks like this.
x square plus y square plus 2gx plus 2fy plus c equal to 0 and in this case the center is half of x coefficient. So remember, what is the center? Center is the coefficient negative of coefficient of x's half and negative of coefficient of y's half.
That is the center. That is the center. And what is that?
Minus g comma minus f. And the radius is g square plus f square minus c. That is the radius of the circle.
So this is the radius of the circle. Then, if I talk about the nature of circle, the radius is root of g square plus f square minus c. If g square plus f square minus c is greater than 0, then this is a real circle with finite radius. Because a finite non-zero radius will come.
And if g square plus f square minus c is 0, then it means radius is 0. So, in this case, we will get a point circle. That is, not a circle, but a point will be found. Because the locus of p will become the center itself. p will coincide with the center. Because the distance is 0, the radius is 0. So where will the point go?
It will be at the centre. So in that case, we will get a point circle. And if G square plus F square minus C is less than 0, then you can see that the radius will be imaginary in this case, and we will get an imaginary circle in this case. So what this means is that we can know from the term G square plus F square minus C that the circle is real with a finite non-zero radius, or it is a point circle, or it is an imaginary circle. Let me tell you one more thing.
If someone says circle is real, What will be the condition of real circle? And real circle condition will be G square plus F square minus C should be greater than equal to 0. Point circle is also a real circle. So what is imaginary circle?
Whose radius becomes imaginary. So this condition is for real circle. Then let us come to intercepts of a circle on X axis and Y axis.
What is the meaning of intercept? Intercept made by the circle on coordinate axis is the distance between the points where the circle intersects y and x axis. For example, this is our y axis and suppose this is x axis and we have a circle and we have this circle and it is making some intercept on it and on this. So this intercept is y intercept from here to here. For example, this is c and this is d.
So this is y intercept. and this is a and this is b, this will be called length x-intercept. This will be x-intercept.
This is x-intercept and that is y-intercept. The intercept made by the circle is the distance between the two points where the circle cuts axis of x and y. x-intercept is root, to root g square minus c.
To root g square minus c. And one more thing, if g square minus c is greater than 0, then obviously, x-intercept is a finite, we will get real, that is, we will get x-intercept, that is, the circle will cut the x-axis at two real and distinct points. If g square minus c becomes zero, then in this case, if it becomes zero, then think when will it become zero? If I take it up, then you can see that the x-intercept is getting smaller and smaller, and how much has become here?
Zero. That is, in that case, what will the circle do? It will touch the x-axis.
Circle will touch x-axis. and if g square minus c less than 0 happens, then the circle will neither touch nor intersect. That means the circle will either be above the x-axis or below the x-axis.
The picture of the circle will look something like this. How will the picture of the circle look? The picture of the circle will look like this. Either it will be above, neither it will touch nor it will cut.
This is your axis of x. It will neither touch nor intersect. the axis. So this is the story of x-intercept.
So we can get a lot of information from x-intercept that circle cuts our x-axis or not, touches or not or neither touches nor cuts x-axis when g square minus c becomes less than 0. Just like this story of y-intercept and y-intercept is 2 root f square minus c and if f square minus c is there then circle always cut the Y axis at two real and distinct points. In this case, if someone says that in this case the photo will be like this and in this case the photo will be like this, you will have a picture of this sort. In this case, the circle will touch the Y axis.
So here in this case, you will have a picture of this sort. This is how our circle will be. This will be the circle.
Or... If it is less than 0, then either our circle will be on the left of the y-axis or on the right of the y-axis. The entire circle will lie either to the left or to the right of the y-axis.
So this is the y-axis. So again, the y-intercept and x-intercept gives you an idea of the location of the circle. And it also tells us that the circle cut karega ya nahi karega x axis ko y axis ko.
Okay. Yaha par agar circle, dhyan rakhiye ga, agar circle x axis ko touch karta hai. If the circle touches x axis.
If the circle touches x axis at say alpha comma zero, then the coordinates of its center will be, that is, if it touches minus g, if I say it touches alpha comma zero, then it will be alpha comma, alpha comma we can say in this case, which will be the radius of the circle. You must have understood that the x coordinate will be the same, because it will be on the same vertical line. So if this alpha comma 0 touches the x-axis, then its x coordinate will be alpha and the y coordinate will be mod of y coordinate.
The y coordinate will be equal to r. If this is the radius, then this will be r plus minus r. Why?
This can be done at the top as well as at the bottom. So remember this thing. Should I say alpha comma r or alpha comma minus r depending on the location of circle.
If the circle is above the x-axis, then when does the circle touch the x-axis? Similarly, if the circle touches the y-axis at beta, then if it touches the y-axis at If it touches the y-axis at, then it touches the y-axis at, and you want to find out the equation of this circle, then its y-coordinate will be beta. Right? and the x coordinate will be plus minus r that means r comma beta or minus r comma beta that will be the coordinate so this is y so remember this that is how you can find out the equation of circle which touches x axis or y axis then diametric form of circle equation of circle with diametric ends x1 y1 and x2 y2 that circle equation okay How do we find out this equation? This is found by using the property of the circle that the angle in a semicircle is 90 degree.
So this point is x1, y1, this is the diametric end and this is the diametric end, x2, y2. And we know that if I add this to the boundary, then what will be this angle? 90 degree. Suppose this is A, this is B. So we can say slope of AP into slope of BP.
is equal to minus 1. And from there, the condition we get is this. That is the equation of circle. x minus x1 into x minus x2 plus y minus y1 into y minus y2 is equal to 0. If you want the smallest circle to pass from two given points, then keep in mind, from two given points, you can always find many circles.
You can find an infinite number of circles passing through two points, two given points. But in all those circles, the smallest circle, like this circle, here, you can have an ample number of circles over here, make as many as you want of circles, which pass through these two points. But in all these, the smallest circle will be the circle which is made with the diameter of these two points. That is, the smallest circle is the one which has been described on this line. as diameter.
The green circle here is the smallest circle for which the line will act like a diameter. So, whenever you get a question that the smallest circle passes from these two given points, then take it from the end of the diameter and write the circle equation. That will be the required circle. Last time we proved this.
Now, let's come to the point of use. Suppose we have two quadratic equations such that the abscissas So, the abscissas are roots of the diametric ends of the abscissas are the roots of abscissas. What are abscissas? They are the roots of abscissas are the roots of diametric ends. What are abscissas?
They are the roots of diametric ends. They are the roots of the diametric ends. What are abscissas? Roots of diametric ends. It is such that the abscissas So, the abscissas are the roots of.
Let's correct it. Abscissas are the roots of. What are abscissas?
Are the roots of. Are the roots of diametric. Are the roots of the diametric ends. Let's write it again. What are these?
These are the roots of diametric ends. Absorbs are the roots of the diametric ends. Abscissas are roots of diametric ends such that we have quadratic equations such that abscissas are roots of diametric ends of first equation. Suppose we have two quadratic equations such that abscissas are the roots of the diametric ends of the first.
equation while the ordinates of the diametric ends are roots of the second equation then the equation of circle with diametric ends can be written as written by adding the two quadratic provided the coefficient of x square y square in Hindi here it is said suppose I have a quadratic its roots are x1 x2 this x1 x2 is the x coordinates of any circle of diameter of any end point that is written in English we have written it like this let us tell you So here Ax square plus Bx plus C equal to 0, its roots are x1, x2 and what is x1, x2? They are the x coordinates of the end points of any circle of any circle. That means its roots are the x coordinates of the end points of its diameter. That means its roots are x1, x2. So this is the end point of this diameter.
And this is... This is the end point of this diameter. Similarly, this is another equation.
Its roots are y1, y2 and these are its diametric ends. These are y coordinates. Now, how will we get the equation of the circle which will be made by taking the diameter of x1, y1 and x2, y2?
You just divide this equation by a and divide it by p. That is, make the coefficient of x2 and y2 as 1. So, we have made the coefficient of x2 and y2 as 1. So, then you have to add both. So, after adding, the resulting equation will be the equation of the circle drawn on x1, y1 and x2, y2 as diameters. This is clear. This equation will be drawn on x1, y1 and x2, y2 as diameters.
Correct spelling of diametric ends. Otherwise, people will not leave this. They will write in comment box.
Sir, spelling is wrong. Very good. Diometric ends.
X coordinate, Y coordinate ends. Y coordinates of diametric ends. Okay.
So, this is done. We will remember this. Then next, position of a point with respect to a circle.
If you want any point in the circle, the point is outside the circle, inside the circle or above the circle, then it is very easy to put that point in the circle equation. When you put that point in the circle equation, that is, put x1, y1 inside s. If it is greater than 0, then the point will be outside. If it is equal to 0, then the point will be on the circle. And if it is less than 0, then inside.
That is, out on greater than 0. On this, on. On this, on. inside the circle. That is how you can know the position of a point with respect to a circle.
Then let's come to the greatest and least distance. If I want the greatest or least distance of any point, P, I have A point, X1, Y1, I want its maximum or minimum distance from the circle. If the point is on the circle, then the minimum distance will be 0 and the maximum distance will be equal to the diameter.
Maximum distance will be equal to the diameter. But if the point is not on the circle, then in this case, this will also be valid. When the point is on the circle or not.
So it is valid for both the cases. So whether the point is on the circle or not, it does not matter. For example, if the point is outside or inside, then the maximum distance will be AC plus R.
That means distance from the center plus R. That will be the maximum distance. So minimum distance will be AC minus R modulus.
AC minus R modulus. Minus R modulus. So whether it is outside the point circle, whether it is inside the point circle, whether it is on the point circle, still these things are valid. Suppose it is on the point circle, then see this.
See this yourself. Some people might be thinking, this is C, this is A. In this case, the maximum distance will be AC plus R.
This will be the maximum distance. And what will be the minimum distance? AC minus R.
AC minus R will be 0. It will come automatically. Clear? So, wherever the point is located, there is no difference. Next, Parametric equation of circle.
Assume that I have a circle x square plus y square equal to r square. So, its parametric equation will be x is equal to r cos theta y equal to r sin theta, where theta will be the angle which will connect the line of origin from O to P with x axis, with positive direction of x axis. So, the range of theta is, In this case, 2π is open from 0. 2π is open because often children ask why 2π is open. 2π is often because 2π has the same coordinates which are at 0. Because cos is 0 and cos is 2π. sin is 0 and sin is 2π.
So, there is no use of writing the same point twice. So, what people did is they put 2π open. Now, in the parametric form, if the circle's centre is h,k, So, in the circle equation, in parametric form, x is equal to h plus r cos theta, y equal to k plus r sin theta. And in this case, center h comma k. So, from here, if I make a line again, passing from center to x axis parallel, then this line, with this line, which will become theta, that will be the theta which will be used over here.
And this will be h plus r cos theta, and in this also, the range of theta will be the same, which we have told there. From 0 to 2 pi k, in between. So, this was the parametric equation of... What is this function?
This function is used when we have to assume a point on the circle. So, the point we assume on the circle is assumed in the form of parametric coordinates. Now, let us talk about tangent and normal.
Tangent is the limiting case of secant. Everyone knows this, it is also taught in calculus. Normal is a line which is perpendicular to the tangent at the point of tangency.
And in the case of a circle, what is always normal always passes through the center. The normal always passes through the center. So, let's assume that I made a line here which is at this point P. And what is this line?
It is perpendicular to this tangent. So, this line will be normal. This is 90 degree. This is tangent.
And this is normal. Because you know that you studied in class 8th. Why will the circle always pass from the center? Because you studied in class 8th.
If we make a tangent at some point on the circle, then where that tangent contacts the circle, the point of contact, if I take that point and connect it to the center, that is, make a radius at that point, then what is this angle? It becomes 90 degrees. This angle becomes 90 degrees. In 7th and 8th class, it is taught somewhere.
So, when we make a line at this point, which will be perpendicular to the tangent, it will definitely pass from the center. That is why all the normals of the circle pass from the center. And one important thing to remember is that if you ever want to find out the relation of line and circle, that how the line behaves in respect to the circle, then you always find the length of perpendicular to the line from the center of the circle.
If the length of perpendicular radius is more than that, then the line will pass outside the circle. Without having any point in common with the circle, the line will pass outside the circle. If P is equal to R, then the line will be tangent to the circle. And if P is less than R, then the line becomes secant.
That is, it will be cut at two distinct points. And if P is zero, then the line will definitely become diameter. That is, if you have a line with a circle...
what is the relation between the two? To find out this, length of perpendicular from center to the line, remove it. Right?
And if it comes in this curve, you can see, something or the other will come out of it. Either greater than r will come, or equal to r will come, or less than r will come, or zero will come. So, you can tell, how is it behaving in respect to line and circle. The second way can be, you can solve line and circle.
From line, y equal to mx plus c, if I have a line, So, suppose I have a line y equal to mx plus c. So, what you will do is, you will put this y value in a circle. And as soon as you put it in the circle equation, you will get a quadratic. Now, if the d of that quadratic is greater than 0, then the line, you know that it has two real and distinct roots, that means the line will cut the circle at two real and distinct points, that means the line will be secant.
And if d is less than 0, So line, imaginary roots will be there, that means that line will be cut anywhere. or not touching the line. Line passes outside the circle.
And if D is zero, then in that case, the line will always be tangent. Be careful. This D thing is true for every second degree curve with respect to a line.
That is, if you want to see a parabola, the line touches the parabola, cuts it, or passes through it, then you solve the line and parabola and find out D. So you can do this. So, this is what happened.
Now, let us come to the forms of tangent. First, this is Cartesian form of tangent. Let us assume that I want the equation of tangent at the point x1, y1 on the circle. So, this is Cartesian form equation of tangent. Equation of tangent at...
P, X1, Y1 on the circle. So, remember, to apply this form of tangent, it is important to be on the circle of that point. Often, children apply the equation of tangent drawn from some point. Whereas, to apply this form, where should that point be?
It should be above the circle. If it is not above the circle, then you cannot apply this form. You cannot apply this form. Remember this.
This form has to be applied when the point is above the circle. If the point is not above the circle, then you cannot apply this form. The miracle result is possible. Parametric form.
Assume I have a circle. x square plus y square equal to a square. And its parametric coordinates are of some point. a cos theta, a sin theta.
Assume I want a tangent at this point. What will happen? How to remember this?
I will tell you that too. How to get tangent? x square is xx1, y square is yy1, 2x is x plus x1, 2y is y plus y1. That is how the equation of tangent is written. Okay.
Suppose I want tangent on 2,3 and 2,3 is on 3, so I will write 2x, 3y plus, now instead of 2x, x plus 2, x plus x1. I want tangent on 2,3. As an example, I am writing tangent on 2,3. and plus 2, 2 will be removed, y, how much to write, y plus f into y plus 3, and plus c is equal to 0, so in this way you can write tangent, in this way you can write tangent, and in parametric form, x cos theta, x cos theta plus y sin theta is equal to a, In parametric form, the tangent will be x cos theta plus y sin theta is equal to a.
So this is the tangent. This is done. Here a will be cancelled out. Basically, what you have to do is, here you have placed a cos theta in place of x1, and y1 in place of a sin theta.
This is x1, this is y1, and you have taken xx1, yy1 in this, because this was the circle. x square is xx1, y square is yy1 is equal to a square. Put it and you get this equation.
You will get this equation at the point p theta. Whereas if this is a circle, then there is no problem if we have a point p theta. p theta is h plus a cos theta, k plus a sin theta.
Assume we have this point. Okay. So in this case, x will be x minus h, y will be y minus k. So you have to remember this. This is the parametric form of tangent and that is the Cartesian form.
Where do we use this parametric form of tangent? Where there are locus problems. We will maximum times use this parametric form of tangent at points in questions where we have a locus problem.
Where there are locus problems. Point of intersection. If I want point of intersection, At A cos alpha, A sin alpha, then the point of intersection of the tangents at P alpha and Q beta, it is given by A cos alpha plus beta by 2, A sin alpha plus beta by 2, divided by cos alpha plus beta by 2, and here also cos alpha plus beta by 2. That means if I have circle x square plus y square equal to A square, and on top of that there are two points P alpha and Q beta so on these two points the tangents will be their Their point of intersection will be this. a cos alpha plus beta by 2, a sin alpha plus beta by 2, upon cos alpha minus beta by 2, into cos, and below also cos alpha minus beta by 2. So this is the coordinates. Keep this in mind.
Then, equation of tangent in slope form. If we talk about slope form, I have a circle x square plus y square equal to a square. tangent of M slope is y equal to mx plus minus a root 1 plus m square.
That means, how many tangents of A slope will there be? Two tangents. That means, on the circle, two parallel tangents of the given slope can be made. And if the circle is this, then you don't have to do anything in it.
Just put y minus k instead of y and x minus h instead of x. Most of the time, this is not used. If you ever want to write tangent on the circle, what will be better? Write y equal to mx plus c. Write...
and length of perpendicular from center is equal to radius. Use this method. You will get the tangent equation smiling.
But still, we have told you this. One more thing I would like to tell you. The tangents, parallel tangents, are always diametrically opposite. Remember, any two tangents which are parallel, any two tangents to a circle which are parallel, they are always diametrically opposite.
opposite to each other. This will always be diametrically opposite to each other. That means the distance between them will be equal to the diameter of the circle. So the distance between two parallel dangers is always equal to the diameter of the circle. So if tomorrow a question like this comes up, suppose you are seeing two parallel dangers, and you are told that these are dangers to some circle, find the radius of the circle.
Now you cannot see this, that how to find radius of circle, it just gives us two lines. If those two lines are parallel and there are tangents on the circle, then the distance between them will be the same diameter and if you divide it by half, you will get radius. So these small points are very useful sometimes.
Next is the director circle of given circle. What is director circle? Director circle is a locus of a point, from which both the tangents are mutually perpendicular to the circle. So it is the locus of a point which moves such that the pair of tangents drawn from it are mutually perpendicular. So if we can find a locus of a point which moves in such a way that the distance between the two tangents is 90 degrees, then that point will move on a locus of a circle.
will be the locus called director circle. Now, because the name itself is director circle, you must have understood that the locus of it in this case will be the director circle. That will be a circle, basically. So, the important thing is that the director circle is always concentric with the given circle. Director circle of a given circle is always concentric with the given circle.
And its radius is our director. circle, C director. And this is our circle, given circle. So this is given circle, this is director circle. So keep in mind, the radius of director circle is root 2 radius of given circle.
And what is its property? If you make any point of this circle, if you make any point of this circle, then they will always be perpendicular to each other. So if I say that I have a given circle x square, x minus h square plus y minus k square is equal to r square. So the equation of C director will be x minus h square plus y minus k square because the center is the same.
And this will be root 2 r square. 2r square. That is the equation of the director circle.
This will be the equation of the director circle. For this circle. Now let's move on to the next topic.
What is common tangent? Common tangent is a line which touches a set of given curves is said to be their common tangent. That means a line which touches a given set of curves, that line is called common tangent.
Those two curves, one circle, one parabola, one hyperbola and one circle, can be called. So, an ellipse and a hyperbola can be any set of curves. In most of our syllabus, the curves are two curves. That means, two circles, one parabola, one circle, one circle, one hyperbola, one ellipse and one circle. So, in this way...
We have a set of curves, and we talk about only two curves in it. So, such a line which touches all the given set of curves, we call such a line as common tangent. For example, these are two circles, and these two circles are touched by this line. So, what will happen?
These two lines are common tangent. What are these two lines on both these circles? Common tangent. The common tangent is of two types.
One is direct common tangent, and the other is internal common tangent. So, direct common tangent. External common tangent or direct common tangent. And this is called internal common tangent.
This is called Direct Common Tangent. This is Internal Common Tangent. In short form, we call it DCT. This is Internal Common Tangent or Transverse Common Tangent.
common tangent which we call as TCT in short form. So, we can make two types of tangent on two curves, one is direct common tangent, DCT and other is TCT. How will we recognize both? If both the centers fly in the same direction in respect to tangent, both the centers but fly in the same direction in respect to the tangent, then those are called direct common tangent or DCT.
And if both the centers fly in different directions in respect to the tangent line, as you can see, one center is above and one center is below. One center is above and the other center is below this line. So this line is transverse common tangent.
So you should understand how to recognize direct common tangent. If both the tangents, If there is a common tangent, then both the centers of both the circles lie on the same side of this line. Then this line is called as direct common tangent or DCT. And if the circles happen to lie on opposite sides of the given line, then this tangent line is called as transverse common tangent.
So if I say this line is L, it is a common tangent. And its center is x1, y1. And its center is x2, y2.
What will happen in direct common tangent? Direct common tangent will be if I put this point L in x1, y1 and along with that, I will put this point x2, y2 in line L. I will put it in the equation.
It is on the same side, so how will it be? Sir, this will be greater than 0. This will be greater than 0. Whereas in this, if suppose this is line L and I will put the centers of both of them and multiply them, then the sign that will come, how will that sign come? That sign will come negative. We had read in straight line that two points in respect to one line lie on the opposite side.
If after putting both the values of the line equation, the values that come after multiplying them, if they are positive, they lie on the same side and if they are negative, they lie on opposite sides. So this is done for TCT and this is done for DCT. So direct common tangent and transverse common tangent. Then comes their length. length of direct common tangent.
So, the distance from here to here, a-b, or this distance, this distance between points of contact, is called the length of DCT. So, the length of DCT will be nothing but the distance between the points of contact. So, this is the distance between the points of contact. B dash, both the lengths are equal. This length and this length are equal.
In the same way, if I look here, here, if this point is C and this point is D, and this point is C dash and this point is D dash, then in this case, the length of tangent, length of TCD, which we call, will be CD, or C dash D dash, that will be the length of tangent. Now, what is this length? So, the length of DCD is d square minus whole square of R1 minus R2.
The length of direct common tangent is nothing but That is nothing but d square. What is d? Distance between the centers of the two circles. d is the distance between the centers of the two circles. d is the center.
d is the distance between the centers of the two circles. And r1 and r2 are the radii of the two circles. So here we have one center and one center. So these are the two centers.
c1 and c2. These are the two centers. And this is the length of C1 and C2.
This is the length of D. What is D? Distance between centers. D is distance between centers. D is equal to distance between centers of two circles.
Distance between centers. Distance between centers of two circles. Centers of two circles. And what is R1? Their radius.
R1 is their... Radius of the two circles, R1 and R2. So the length of the direct common tangent is given by D minus R1 minus R2 ka whole square. Aur ek cheez yaad rakhiye ka bahut important property hai, ke agar aap DCT ko ka jo DCTs hain, un dono ka point of intersection dekhen, if you look at the point of intersection of DCTs, then the point of intersection of the DCTs will be such that, This is an important property. Will be such that they divide the line joining the centers of the two circles externally.
In the, this and this. Line joining the centers of the two circles externally. Assume this point. This is the point, this is P and this radius is R1 and this radius is R2. So, keep in mind, PC1 ratio PC2, PC1 ratio PC2 is nothing but R1 ratio R2.
That is the point of intersection of the direct common tangents divide the line joining the centers externally in the ratio of their radii. So, this thing important. It divides it externally in the ratio of their radii. That means PC1 ratio PC2 will be R1 ratio R2.
In the same way, if both the circles have equal radius, then in that case DCT will be parallel. And in that case their point of intersection will never come. Because the lines will be parallel. They will meet at infinity as they say.
If we talk about TCT, then in TCT, if we talk of TCT, then in case of TCT, So, that is why you must have understood that they are also called external common tangent. So, the main reason for calling them external common tangent was because they divide the line joining the centers of the two circles externally in the ratio of the radii. Their name is also internal common tangent because they divide the line joining the centers of the two circles internally in the ratio of the radii.
So, suppose here these are the centers. These are the centers of the two circles. This is C1 and this is C2. And this point is Q. So, here what will happen?
The line joining the point of intersection of the two TCTs divide the line joining the centers of the two circles internally in the ratio of the radii. This is R1 and this is R2. So we can say QC1 ratio QC2 will be equal to R1 ratio R2. And its length of TCT is D minus R1 plus R2 whole square. And you can see here TCT will be smaller as compared to DCT.
That means the length of direct common tangent is always greater in respect to TCT. It is greater than the length of TCT. So length of DCT is greater than the length of TCT. So remember this thing. And the point of intersection of internal common tangent divides the line of both the centers internally in the ratio R1 is to R2.
And the point of intersection of external common tangent divides the line joining the centers of the two circles externally in the ratio R1 is to R2. So that is the thing we have written here. The point of intersection of internal common tangents, that is TCT, divides the line joining the centers of two circles. internally in the ratio of their radii or here externally in the ratio of their radii. Radii.
Then comes that the number of common tangents is decided according to the position of two circles. How? Suppose both the circles are separated. The two circles are separated. How are both the circles separated?
That means both do not intersect each other. Or you can say that both do not have any common point. So in this case, what is the condition? R1 plus R2 is less than C1 C2.
That means C1 C2, which is this line, its length will always be more than R1 plus R2. You can see that this is R1 and this is R2. The length of C1 and C2 will be more than this.
And in this case, what will happen? There will be four common tangents. We will have four common tangents in this case. How? How will there be four common tangents?
Sir, two will become DCT. Two will become DCT and two will become TCT. So two DCT and two TCT. Two direct common tangents and two transverse common tangents. So we will have Two transverse common tangent.
Then after this, if both the circuits touch each other, then what will be the condition for that? Distance between the centers should be equal to sum of their radii. C1, C2 should be equal to R1 and R2's sum. And in this case, one became TCT. Why TCT?
See, both the centers are on different sides. This became TCT. And the DCT will be these two.
These two will be. So, two is DCT. This will have 2 DCT and plus 1 TCT. This is TCT and this is 1 DCT and this is also 1 DCT.
And the condition for this is C1 C2 is equal to R1 plus R2. In the third case, if both the circles intersect, so if the two circles intersect, then obviously the distance between the centers is lesser than the sum of their radii. C1 C2 is less than R1 plus R2.
R2. But along with that, one more condition is required. It is greater than mod of R1 minus R2.
It should be greater than R1 minus R2. Sir, why should it be greater? You have read this in the class. In the comment box below, tell me the reason for this. Why should T1 C2 be greater than R1 minus R2?
Why should it be greater than R1 minus R2? So, in this case, two tangents are formed and both of them are DCT. Two DCT are formed. You will have... two DCTs in this case.
So you will have two DCTs in this case. Then, if the circles happen to touch internally, then you see carefully, these two circles are touching each other internally. Suppose this line is C1 and this center is C2. So C1, C2 will be R1 minus R2.
So, the value of C1, C2 will be equal to R1 minus R2k. We will put a mod because we do not know which circle has a larger radius and which has a smaller radius. However, in this diagram, we can clearly see that R1 is bigger than R2. So, the larger the radius, the smaller the radius is equal to R1. By reducing the smaller radius in the larger radius, we will keep it equal to the distance between the centers.
Then, this is the condition in which the two circles will touch each other. So, if two circles are touching each other, internally touching each other, we will have a larger radius. So in that case distance between the centers should be equal to difference of their radii. And in this case you can understand that only one tangent will be formed and what will be that tangent? So both the centers are on the same side.
Only one tangent will be formed and that will definitely be DCT. Only one DCT will be formed. So this should be equal to R1 minus R2.
Now maybe you have understood that why C1 C2 will be greater than R1 minus R2. Because if C1 C2 is greater than R1 minus R2. R1 minus R2 will happen or not then in that case that condition can also come when one circle will enter the other circle so C1 C2 less than R1 plus R2 should not happen it should be bigger than R1 minus R2 because because of being bigger than R1 minus R2 this guarantee is given that one circle will not enter the other circle but some part will remain outside that means figure will be of this kind ok Then after fourth, fifth, if a circle is completely contained, any distance between the centers is less than the difference of the radii, then in that case, no common tangent.
You will have zero common tangent. In J-Mains, there are many questions on this, that two common tangents can be made on these circles, four common tangents can be made on these circles, and in which circle, lambda, lambda, put the coefficient of x, you are asked to find the set of values of lambda. This kind of questions are asked in 2024, 2023, and 2020. So, remember these are important conditions. You may get to see this question in your exam.
Then equation of chord joining, two points on the circle, x square plus y square equal to a square. Assume we have a circle and above it, this circle is in standard form, i.e. centered at 0, 0. So this circle is centered at 0, 0. And we want to find out the equation of, this is the center, the center is 0, 0. And we want to find out the equation of the chord joining to x square plus y square. points on this circle and the points are p alpha and q beta say the points are p alpha and q beta these are two points we need the equation of the line of chord that connects these two points so the equation of chord is x cos alpha plus beta by 2 plus y sin alpha plus beta by 2 is equal to a cos alpha minus beta by 2 and you know what alpha and beta are basically this is what we have see This is what we have read in the parametric form.
So, the alpha and beta are these angles. And I hope you appreciate that. This will be beta and this angle will be alpha. This is beta and this angle is alpha.
So, the equation for the chord that will connect these two points will be this. Then next, length of tangent. Length of tangent, suppose P is an external point and If we make a tangent on any circle, the length of the answer is, let's say it touches A and B. The distance from P to A or P to B, which will be equal to P A is equal to P B. So these distances are called the length of tangent. The length of tangent is P A as well as P B. So the length, this is P A. Or you can say it is equal to P B. So, PA and PB that is equal to root S1.
Root S1 means, put that point in the circle equation, put the root and you will get the length of the tangent. So, the length of the tangent drawn from any point to pass, what should that be? That will be root S1.
That point should be put in the circle equation. You will get the length of tangent drawn from X1, Y1 to the circle. So, a given circle is this.
Here, in the circle equation, we have called x1, y1 as s1. So, in s, we have x1, y1. Here x1, here y1, here x1, here y1. And here, we have c. So, we will put the value of this, the root.
That is called as length of tangent. To find the length of tangent, remember one thing. Here, the coefficient of x2 and y2 is unity. If unity means 1. If it is not 1, then we should make 1 first.
and then only after that, the length of tangent should be taken out. If tomorrow we have 2x square plus 2y square, then first you will divide the whole equation by 2, so that the coefficient of x square and y square becomes 1, only then you will find the length of tangent by using this formula. So remember this. Then comes the chord of contact. Chord of contact means, we can always make two tangents at an external point in the circle.
So if we make two tangents at the circle, wherever it touches the circle, The two points A and B, the line that connects these two, we call it chord of contact, i.e. COC. And its equation is the same, t equal to 0. t equal to 0, i.e. as you had written tangent, write the same expression again.
In which respect? In p's respect. i.e. if I have circle x square plus y square plus 2gx plus 2fy plus t equal to 0. So COC, i.e. cord of contact, which will be AB, its equation will be xx1 plus yy1 plus gx plus x1 plus fy plus y1 or plus c is equal to 0. So that will be the equation of tangent. So tangent means tangent equation.
But who will this equation be given? It will be given as the chord of contact. What is the difference between this and tangent? Listen carefully.
In tangent, this point P was above the circle. But in this case, this point is not above the circle. It lies outside the circle.
So, if the point lies outside the circle and you form a similar expression as for the tangent in this case, then in this case, this equation, this expression equal to 0 will give you what? It will give you the equation of chord of contact. not the tangent and you can also understand if this point can be pulled slowly where it should come it should come on this circle so in that case the chord of contact will be tangent.
Why? If the point is close to the circle, then the chord of contact will break. Chord of contact will become smaller and smaller. And as soon as point P will come on the boundary of the circle, then the chord of contact will also come on the boundary and it will become tangent.
So, the expression of this is the same. It is a tangent expression, but the equation is different. Equation different means the equation given in this case is chord of contact, not tangent.
What is the difference between the two? The point was on the circle there, and the point was not on the circle here. So this is important, this is to be noted.
We will remember this thing. Then chord with given midpoint. If the chord is given, if its midpoint is given, then there is only one chord in the world passing from that point. First thing to remember is that if the circle is given and it can have many chords.
But if you want a chord whose midpoint is a given point, then there will be only one chord in this world. That means, if this point is M, this is 90 degree, that means this is the midpoint of this chord, then this chord will be unique. That means, there will be countless chords passing through M.
There will be infinitely many chords which pass through M. But there will be only one chord for which this M shall act like. shall act like midpoint. Aisi sirf ek hi chord hogi.
Aur wo chord kya hogi? A, B. Toh yaani ke, agar chord ka midpoint de diya jaye, toh aisi kitni chords hongi, jiske liye midpoint act karega? Aisi duniya mein sirf ek hi chord hogi. There will be only one chord, for which this will act like the midpoint. Aisi duniya mein sirf ek hi chord hogi.
Aur koi chord nahi ho sakte. Aur iski equation kya hoti hai? T barabar.
S1, T equal to S1. Do you know what is the best thing? If tomorrow there is a parabola, you can make two tangents from the external point of the parabola.
Its chord of contact equation will be T equal to 0. If a chord of a parabola is given as the midpoint, then such a chord will again be unique in that case. Its equation will also be T equal to S1. That means, what we are reading, that the chord of contact equation is T equal to 0. And then, chord with given midpoint, T equals S1.
All these things are the same in conics. In circle, ellipse, parabola and hyperbola. They are the same. There is no difference. So, T equals S1.
So, in my opinion, you understand. What do we do for T? What do we do for T?
For T, you will remember. Let me remind you again. By the way, I have told you.
I will write you in place of x square, xx1. In place of y square, yy1. In place of 2x, x plus x1.
Or... Instead of y2y, y plus y1. y plus y1.
That is what we do for t. And what do we do for s1? For s1, put x1, y1 in equation of curve. In equation of curve. In equation of curve.
And in this case, what is that curve? That is circle. Keep that point in the circle equation. x1, y1.
What is x1, y1? x1, y1 is midpoint. Suppose we have written midpoint P here.
And coordinates of P are x1, y1. So we can write its equation with t equal to s1. Caught with given midpoint. And as I told you, tomorrow if parabola comes, in that also caught with given midpoint, equation t equal to s1. Hyperbola comes, t equal to s1.
And how will we write t? In the case of parabola, y square instead of yy1. If x square, then instead of x, x1. And 2x instead of x plus x1.
Or t. and instead of 2y, y plus y1. This is what we can do in hyperbola, in ellipse, in everything. Okay?
Equation of pair of tangents to a given circle. Let's assume we have an external point, and we want to find the equation of tangents drawn from an external point. Let's assume it's an external point, and from here we want to find the joint equation of both the tangents. We saw the pair of straight lines, read them. So, now we are going to talk about the joint equation of PA and PB.
So, the joint equation of PA and PB is, suppose this point is P and this point is X1, Y1. So, joint equation of PA and PB is given by SS1 is equal to T square. SS1 is equal to T square.
What is S? S is the equation of the given curve. As it is written in X, just stick it. X square plus Y square plus 3X plus 4Y plus 5. Suppose it is written here, then it will be S.
And what will be S1? which I just told you, put the point in the circle equation. And what will be T?
This is done. So SS1 is equal to T square will give you the equation of pair of tangents to the circle drawn from an point drawn from a point Px1y1. Now let's talk about family of circles. Family of circles, family of curves, what does it mean?
Family of curves, any group of curves, family is formed when they share at least one common property. A group of curves which share at least one common property will form a family of curves, of that curves. It doesn't mean that you have to make 10 lakhs or 1000s or 10 crores or 1 crore. curves and you made them family. No.
Condition is they should share at least one common property. At least one property should be there which is common for all of them. Which all follow.
Like here equation of family of circles which pass through the point of intersection of two circles S1 and S2. Assume we have two circles. S1 and one circle is S2.
So we have two circles. One circle is S1 and second circle is S2. Now from the intersection point of these two pass hone wale, itne circle ban sakte hai?
We will have infinitely many circles passing through their point of intersection. Jaise ke agar mai kahun, say this is one circle, say this is one circle which passes through their point of intersection. Ek circle ye ho gaya.
Now ek circle hum yaha bana sakte hai. Ye jo circle hoga, ye hai bhi in dono ke point of intersection se jaayega. This is the circle which passes through their point of intersection. How many circles can you make by doing this?
Infinitely many circles can be drawn which pass through their intersection. I have made some circles here. This circle is S1 equal to 0. S1 is its equation. And this circle is S2 equal to 0. This circle is S2 equal to 0. So all the circles which will go from the intersection points of these two, intersection points meaning those which will go from A and B, all the circles will be their equation will be S1 plus lambda S2 equal to 0. S1 plus lambda S2 equal to 0. Here lambda is a parameter and lambda will not be equal to minus 1. Because if lambda will be equal to minus 1, then in that case, X square will be eliminated. This will become a straight line.
In that case, you will not get a circle but a line. Which we will see now, it becomes a radical X. So, that is why...
To write the whole family of circles, that is, to write all those circles that pass through these two points, that family of circles is S1 plus lambda S2. Here, one thing to remember is that S1 plus lambda S2, this lambda will be unique for every family member. That is, suppose this is a circle, we are talking about this circle, so for it, lambda will be 1 or 2. If this is a circle, then for it, lambda will be something else.
It cannot be that one... One family member will be given one value of lambda. And every family member will have some unique value of lambda.
So every member of the family corresponds to one unique value of lambda. So every family member, every family means, you will understand, in this whole family of circles, which passes through the intersection points of S1 and S2, in this whole family, there will be infinitely many members. And every member will have a unique value of lambda, which will be different from the others.
If I say, this lambda acts like the average number of each member. So, like every member has a different average number, similarly, every member's parametric value, this lambda, is also different. So, every value of lambda corresponds to only one member of the family.
By the way, family of circles is not in J.E. Mace's syllabus, but it is in J.E. Advance's syllabus.
Then, family of circles passes through two... Passes to the point of intersection of the line S equal to 0 and L equal to 0. Assume this line is L equal to 0 and this circle is S equal to 0. From these two intersections, i.e. from point A and point B, the equation of all the passing circles, let's talk about the equation of those, how many circles will pass? So remember one thing, from those points, from those given points, How many circles can be passed?
There can be infinitely many circles which pass through two given points. Two given points can pass infinitely many circles. You can draw infinitely many circles which pass through two given points. The two points which pass through can be infinitely many circles.
But passing through three points, passing through non-collinear points, there is only one circle in the world. So through two given points... We can have infinitely many circles. But through three given non-collinear points, you can always have a unique circle. So in this case, we will get many circles.
This is also a complete family. And what is the common property in these? As I told you, the common property is that they all pass through A and B. Here also, all the circles that will be formed, they will also have common property. What is that?
That they pass through the intersection of these two. They pass through the intersection of the two circles S1 and S2. In this case also, the common property is that they pass through A and B. And what is their equation?
S plus lambda times L is equal to 0. Here again what happened is that a parameter happened and again there is no need to say that. In this family there will be many circles, infinitely many. But every unique value, every circle will have a unique value of lambda.
And every value of lambda will correspond to some circle of the family. So you should remember this. Then next is equation of family of curves which passes through two given points.
Now we need a circle passing through two given points. So if I need the equation of family of circles which pass through two given points, this point is A, this is x1, y1 and this point is B which is x2, y2. And we need the equation of circles which pass through the points A and B.
So you know how many points there will be like this, how many circles there will be like this, how many circles there will be like this. So how do we write this family equation? We write the line equation of AB passing from A to B.
Let's assume this line is LAB. What is this line? LAB. So this line is LAB. And then we write a circle with AB as diameter.
We write down a circle with AB as diameter. What is this? CAB or SAB.
It is the equation of circle which which has diametric ends as a and b. And this line is LAB, which passes through a and b. So, what will be the equation of this circle? The equation will be x minus x1 into x minus x2, then plus y minus y1 into y minus y2 is equal to 0. And what will be the equation of line AB? The equation of line AB will be y minus y1, is equal to y2 minus y1 upon x2 minus x1 into x minus x1.
That will be the equation of the line AB. And the equation of all the circles which pass through the intersection, which pass through the intersection, which pass through A and B, which pass through A and B, all those circles, all those circles, all those equations, all those equations, I can write them. by using the family which we talked just a few minutes ago. Which family am I talking about? Its S plus lambda L.
Because this is the family of all the circles which pass through the point of intersection of this line and this circle. So, all the circles which pass through the point of intersection of this line are S plus lambda L equal to 0. So, if we see, SAB and L's point of intersection are the only ones which pass through A and B. Because A and B are their point of intersection. So, its equation will be SAB plus lambda AB LAB is equal to 0. So, this will be the equation of all the circles. The equation of the entire circle family will be there.
Then comes the equation of family of circles which touch a given line at the point X1, Y1. Suppose I have a line and we want the equation of all those circles. which touch this line at x1, y1. So we need the equation of all such circles.
For example, this circle here. This line touches x1, y1. A circle can be here also.
So how many such circles will there be? We will have infinitely many such circles. For example, this circle here.
This circle here. Then this circle can also be formed here. So by doing this, how many circles will come?
Infinitely many circles. And this line is L. L means we have its equation. What is the equation?
Sir, L is Ax plus By plus this expression. This is L. And this equal to 0 will be the equation of line.
And this equation of touching all the circles on x1, y1, the equation of the whole family will be Write point circle. x minus x1 square plus y minus y1 square equal to 0. So, we have written a point circle. A point circle.
You must remember, I told you in the beginning, what is a point circle? A circle of 0 radius. X minus X1 square plus Y minus Y1 square is equal to 0. We have written this circle and we have the line.
So, the equation of this entire family will be this point circle plus lambda times L is equal to 0. That will give you the equation of the entire family of circles which pass through, which touch this line. at x1, y1. Because how many circles can be made which will touch this line at x1, y1? We can have infinitely many such circles which touch this line at x1, y1. And that is the equation.
We have done a lot of problems in the class. If you want, you can practice the class illustrations, the PYQs after watching this lecture. Then the whole chapter will be set in your mind and it will be set well. And the revision is going on.
Those who have read the circles, you can say... All the homework that is done in the class which is given by the teachers or those who have tried the modules or wherever they have tried the assignments, this is a very good mind map for them where the entire theory of the circles is being kept in front of you. Next is radical axis. Now what do we mean by radical axis?
Radical axis of two circles. So radical axis of two circles S1 and S2 is the locus of the point which moves such that its power with respect to two given circles are equal. That means in respect to two given circles, its power is equal.
Now what is power? What is power? Two circles are equal.
Power, what is power? Power is basically, if you take a point P and draw a secant or a chord passing You have taken a point P and from here this is B. You have drawn a line from P which cuts the circle to A and B. So this is the distance of PA from this point to the first point of intersection into PB.
This distance is called power of P. Power of P with respect to circle. So what is power? The power of P. P, power of a point P is nothing but the length of the, is the product of the length of the intercepted portion between the point of intersection of the line through P from P.
Kehane ka matla, phir se bol deta hu, ke agar aapne P ek point liya, wahan se aapne ek line banai, jo circle ko A par aur B par cut karti hai, to unke points of intersection ki doori kahan se le le, P se le le, aur in dono ko multiply kar de, to jo aata hai, wo power of point P ke lata hai. Is it necessary to have P external? No, P can be inside as well. P can be anywhere, even on the boundary.
For example, if P is here, if you make a line from P, then this is A and this is B. So, PA into PB, what will this become? This will become, PA into PB will be nothing but this will be the power of point.
So, if there are P coordinates, if P is power of P, Power of P with respect to circle, power of P, x1, y1. What are the coordinates of P? x1, y1. x1, y1 with respect to circle, S equal to 0 is, put this x1, y1 in this circle, is S of x1, y1. Meaning que, S1.
So power S1. We called this S1. So power is the product of distances.
And whose product is this? Put that point in the circle equation. The answer will be P.
Power will be called in this circle's respect. And just now we talked about length of tangent. What do we do in length of tangent?
We put that point in the circle equation. We put that point in the circle equation. And whatever expression comes, then what do we apply?
We apply root. That means if the point P lies outside the circle, then length of tangent is nothing but the root of the power of the point. That means in that case, if the power of point P is applied, if the root is applied on it, then that will give you the length of tangent. If point P is external, then if the root is applied on the power, that will give you the length of tangent in this case. But for power, point P can be inside, outside, in fact on the circle.
So if point P is inside the circle, then root S1 will not give length of tangent, rather it will not be defined in real number. And if point P is on the circle, then in that case, the power will be 0. Because if point P is above the circle, then in that case, P and A will be the same. P will be equal to, so PA into PB, so the length of PA will be 0. and into P also, so negative 0 will be obtained in this case. In this case, how will the power come? Positive.
The power will come positive and when we put its root, we will get the length of tangent drawn from P. That is, if I put the root of power in this case, then I will get this length Pt. That is, Pt equal to root S1.
But what is the power? I can say that Pt square is equal to S1. That is, the length of tangent of Pt is equal to S1. And S1 is equal to Pa into Pb. You may have read this theorem in class 10. If we make secant from any point, then Pa into Pb is equal to Pt square.
That is proved here as well. Rest, we have explained all these things very well in class. So, you should understand the power of power.
What is the power? Take a point. Make a line, P A into P B, so that gives you power.
And how do you get its value? S1. Take that point and put it in the circle equation.
If the point is outside, power is positive. If the point is inside, then you will find power will be negative. Why will power be negative?
Write it down in the comment box and tell me. Why will power be positive here? Write it down in the comment box and tell me. Power will be zero in this case.
I have already told you that. So why will power be positive and why will it be negative? Write in the comment box below.
So for now, if a point P goes in such a way that its powers are equal in respect to two circles, then its locus will be called as the radical axis. So the radical axis is in respect to two circles. A locus of such a point whose power is equal in respect to two circles. Radical axis is a locus of such a point, from which the powers are equal in respect to both the circles.
If we make it here, then P is equal to P. It should be like this. The powers should be equal. If this is A1, then this is B1. If this is A2, then this is B2.
So, PA1 into PB1 is equal to PA2 into PB2. So, a point which satisfies this relation, the locus of that point will be called radical axis. And the equation of radical axis is to minus both the circles.
Just now, I told you in family of circles, S1 plus lambda S2 was the equation of family of circles passing through the intersection of these two points. But, if we keep lambda as minus 1, So in that case, S1 minus S2 will come, which will be a straight line because x square and y square will be cut. And the line which will come to us, what will happen? Radical axis will be formed.
So how do we get the equation of radical axis? S1 minus S2 is equal to 0. That is the radical axis. So that is how you get the radical axis. Here, if two circles intersect, radical axis is the common chord. If two circles intersect, So, from S1 minus S2, we get a locus, a straight line, and that will be a radical axis.
And in this case, the radical axis becomes a common chord. If two circles intersect each other. So, if two circles intersect, then the radical axis is the common chord.
Radical axis becomes a common chord. Why? Because this is the point. Its power in respect to this circle and its power in respect to this circle will be zero.
Why? This point is on the circle. Similarly, the power of this point in respect to this circle and in respect to this circle will be zero. That means, these two points are such points whose powers are equal in respect to both the circles.
That means, the radical axis will definitely pass from this point and from this point. Because all the points on the radical axis lie, whose powers are the same in respect to both the circles. And because the powers of both these points are the same in respect to the circle, then definitely the radical axis will pass from these two points. So, this is the radical axis, which is nothing but the common chord. If two circles touch, then the radical axis in that case...
That will be the common tangent. So if two circles touch each other externally or internally, then in both the cases, the radical axis becomes their common tangent. Radical axis becomes the common tangent.
This is an important point. Radical axis is always perpendicular to the line joining the centers of the two circles. The line joining the centers of the two circles is always perpendicular to the line.
This is always the case with radical axis. You know the equation of radical axis, I told you, S1 minus S2 is equal to 0. So when you write the equation of radical axis, then see, the equation of radical axis, X square is cut from X square, Y square is cut from Y square, 2G1 minus G2 into X2, 2F1 minus F2 into Y, C1 minus C2, that will be the equation of radical axis. And if you see its slope, then we will write slope, this upon G1 minus G2.
g1 minus g2, plus 2f1 minus f2, 2f1 minus f2. So, the slope of this will be g1, g1 minus g2, and 2f1 minus f2. So, when we will see the slope, this upon this with a negative sign.
And when we will see the slope of this, we will multiply both of them. So, you will get the answer, minus 1. Do it now. Do it now.
Write the slope of this and... Write C1, C2 slope and see. And multiply both the slopes. See if you get a minus or not.
From there you will understand that radical axis is perpendicular to the line joining the center of the two circles. It is perpendicular to this. Then next, radical axis need not always pass through the midpoint of the line joining the center of the two circles. Some people think that radical axis is where both the circles join the center of the line, it goes right in the middle.
The C1, C2 have a perpendicular bisector. This is not necessary. Don't think that if I call this point M, then C1M will be equal to C2M. It is not necessary.
This can happen only when both the circles have equal radius. So if the two circles have equal radius, only then the radical axis will be the perpendicular bisector of the line joining the center of the two circles. Then the fifth point, which has been a problem in advance, Radical axis bisects the common tangent between the two circles.
If there are two circles, we just studied the common tangent, then if there are two circles, then the common tangent of both of them, whether it is a direct common tangent or a transverse common tangent, bisects both the tangents. Who? Radical axis. Radical axis bisects the common tangent of the two circles. Suppose this is the common tangent of the two circles.
And suppose this is the radical axis. This radical axis will be bisecting both these parts. This and this. This and this. They will be equal.
Suppose this is point A. This is point B. This is point C.
And this is point D. So, this length is x and this is z. So, Cz will be Zd equal to Zd. Ax will be equal to xb. So, this is a direct common tangent and this is a transverse common tangent.
Both are bisected. No matter which tangent pair it is. Pairs of circles which do not have radical axis are concentric.
If the center of two circles is same, then their radical axis will not exist. Because when we will minus both, So everything will be cut off, c1 minus c2 equal to 0, which is not possible. That is, their radical axis will not be possible. So two such sets of circles, two circles whose radical axis is not possible, are concentric circles.
If one circle is contained inside the other, then the radical axis passes outside both the circles. That is, if one circle is contained inside the other circle, and both are not concentric, then the radical axis of both of them is passes through the exterior of both the circles. Then the radical axis of both the circles will pass through outside of both the circles. That means the radical axis will pass through like this. If it is concentric, then the radical axis will not be there.
But if it is contained in one circle and the other circle, then the radical axis will pass through both the circles. It is never possible that the radical axis will cut and go. A very important property of radical axis can also come in use here.
If I ever have to show that two circles touch each other. So, touch, they may touch externally or internally. For external touch, C1, C2 is equal to R1 plus R2. For internal touch, C1, C2 is equal to mod of R1 minus R2. These were the conditions for external touch and internal touch.
But if I am only interested that two circuits touch, I don't know. Should they touch externally or internally? But I am only concerned they should touch. So, if two circuits touch, then we can use this point. Which point can we use?
This point. If we show that two circles'radical axis touches one of the circles, then definitely it will touch the other circle. It will also touch the other circle.
Hence, the two circles will touch each other. So if you want to show that two circles touch each other, then in that case you should show, and you are not interested whether it will touch internally or externally, then you should show that the radical axis touches any one of them. means if you want to show that two circles touch and you don't care about this at all internal touch or external we just want to see if they touch or not so what you do is take out the radical axis and see if one of the radical axis is touching one of the circles so definitely both the circles both the circles will touch each other at the same point where the radical axis was touching one of the circles ok, so this is done now let's move forward now let's talk about radical radical center key to radical center kia hota hai mali ja maare paas teen circles hain aur teen ho ke centers kiaise hai non collinear suppose we have three circles centers of which are non collinear toh in teen ho ki radical axis pair wise yaani ki s1 s2 ki radical axis jaise yeh hogai maan liye radical axis hai r12 issi tarah se yeh second first circle aur third circle ki maan liye yeh hai r31 aur in dono circles ki radical axis yeh hai r2 and then three.
These three radical axes are concurrent. So if you have three circles whose centers are non-collinear, not in one line, then it is seen that, in fact, it is also proved in class, that if we take the pairwise of these three circles, what should we take? We should take the radical axes. These three radical axes pass through the same point. The point we call radical center.
The common point of intersection of the radical axis of three circles taken two at a time is called radical center of the three circles. So this point is called radical center. So remember this thing. What is radical center?
There are three circles whose centers are not collinear. Because if the centers are collinear, then you can understand one thing. If the centers are collinear, if the centers lie on a straight line, then in this case, the radical axis will come. Because the center joining line is always perpendicular.
So, whatever radical axes will come, they will come parallel to each other. So, they will never intersect with each other. So, in that case, the radical center will not exist. So, it will only exist when the centers of all three are non-collinear.
Then we will get the radical center. If I make circles with the diameter of all three sides of any triangle, then how many circles will be made? Three circles will be made. Now, these three circles will also have radical axis pairwise. And when those radical axis will be there, then we will get the corresponding radical center.
So, in this case, it is seen that the radical center is the orthocenter of the circle. Radical center becomes the orthocenter of the circle. Because the third circle will be made, just look carefully, the third circle will be made, taking the diameter to the sides, it will be something like this.
So, this is the third circle. This will be the third circle which we are making with the diameter. This is the third circle which is drawn on the sides as diameter. You can see three circles.
One circle is this which is made with the diameter. One circle is this which is made with the diameter. And one circle is this which is made with the diameter of BC. So, all three circles are made. Now, the center of all three circles, obviously, where does it lie?
Its center will lie on AB. This one. AB has a diameter. Its centre will lie on AC and its centre will lie on BC.
So, this is absolutely certain that how will the centres of all three be non-collinear. So, definitely, the radical centre of all three will also come. So, the radical centre that comes here is the orthocentre of the triangle.
So, in this case, if we draw three circles taking the sides of the triangle as diameters, then the radical centre of the three circles is nothing but the orthocentre of the triangle. I think you should understand this because both these circles are intersecting here. Where are they intersecting?
Both of them are intersecting here. This is the AC with its diameter. If I say this is SAB and this is...
SAB and SAC are intersecting at D and their point of intersection is A. So, we know that the point of intersection of A and D is a common chord if both the circles intersect. We had just studied this earlier.
So, their radical axis will go from A and D. So, the radical axis will pass through A as well as D. So, the radical axis of SAB and SAC will be that becomes AD.
What is that? AD. Now look carefully, SAB has this diameter.
Who? AB. And this point is D, on its circumference. So how much will this angle be? This angle will be 90 degree.
That means, the radical axis of these two is perpendicular to BC. So this means, this radical axis will be along with altitude. Altitude means, on the opposite side of this vertex, The perpendicular is called altitude. So, from this point, the perpendicular on the front side is called radical axis. So, radical axis is along altitude of the triangle.
So, this is along altitude of triangle. So, these two radical axes are AD altitude. Similarly, these two radical axes are radical axis of SAB and SBC will be nothing but The altitude AE. And in the same way, the radical axis of SAC and SAC. And along with that, the radical axis of SBC will become which side?
It will become CF. And where will all three of these be? They will be along the altitudes of the triangle. And along the altitudes of the triangle, the point of intersection of these three altitudes will be the orthocenter. but this radical axis is also c-calong so their point of intersection will be radical center so it means that radical center will become orthocenter so the point of intersection of radical axis of all three is this point but if we see this is also the point of intersection of altitudes so this is the orthocenter which will become radical center so in total we got to see that if I make a circle by taking the diameter of all three sides then the center of all three circles is non-collinear.
That is why the pairwise radical axis of all three will be at the same point. And the point here is the orthocenter of this triangle. It can be used for a small objective. Now let's talk about the angle of intersection of two circles.
If I have two circles of... If the angle of intersection is 90 degree, then we will say that both the circles are orthogonal. First, we will see how we can write the angle of intersection of two circles.
Suppose you have two circles. How do we find the intersection angle of these two circles? We have made this tangent on the first circle and this tangent on the second circle. This is tangent at the point of intersection.
We have drawn a tangent at this point. Suppose this angle is theta. So this angle theta is called as the angle of intersection of the two circles. So the angle of intersection of the two circles is called as angle of intersection of the two circles.
If this is angle, what will happen? It will be 90 degrees. It means that at the point of intersection, the tangent which was formed on the first circle and the second circle, should be perpendicular to each other.
In that case, we say that both the circles will be orthogonal. So, it should be 90 degrees. So, here it is written that the angle of intersection of two circles is the right angle.
Then, the circles are said to be orthogonal. Now, think about it. If both the circles are perpendicular to each other, then in that case, In that case, it is If it becomes perpendicular, suppose this is the tangent and this is perpendicular to it.
So, obviously, this tangent, both the tangents are perpendicular to each other. The tangents of the two circles are perpendicular to each other at the point of intersection. Suppose these are perpendicular to each other. What is this angle?
Suppose it is a 90 degree angle. So, it is obvious that this will be from its center. This will pass through the center of the first circle.
And this will pass through the center of the second circle. These two will pass through each other's center. Why will they pass? Tell me in the comment box below. Tell me in the comment box below.
What is this angle? 90 degrees. Both of them will pass through the centers of the other circle. That means, the tangent on this will pass through its center, the tangent on this will pass through its center.
Its radius, let's assume, is r1, its radius is r2, and this distance is d. So, what will be the condition? r1 square plus r2 square is equal to d square, Pythagoras theorem. r1, let's assume, the first circle is x1, x square plus y square plus 2g1. x plus 2 f1 y plus c1 is equal to 0. And what is the second circle?
x square plus y square plus 2 g2 x plus 2 f2 y plus c2 is equal to 0. So if these two circles are orthogonal to each other, then what will be r1 square? Sir, g1 square plus f1 square plus minus c1. minus c1 and g2 square plus f2 square minus c2 is equal to d1 distance between the center center minus g1 minus f1 and its center will be minus g2 minus f2 if I write d1 d square then it will be g1 minus g2 whole square distance formula and plus f1 minus f2 whole square and as soon as I open it So, g1 square, g1 square cut jayega.
f1 square, f1 square cut jayega. g2, f2 square cut jayega. Yeh bache ga minus c1 minus c2. Minus 2 g1 g2. Or minus 2 f1 f2.
Dikey. Toh yaha se aage aapke paas 2 g1 g2 plus 2 f1 f2 is equal to c1 plus c2. So, that is the condition for orthogonality.
If this condition is satisfied, then both the circles will be orthogonal to each other. If this condition is satisfied. What is the importance of the radical center that we just studied?
Because we have seen the radical center. What is its use? Before seeing its use, let me tell you one more thing. Suppose you have two circles.
Suppose you have two circles. And all the circles which are cut in the same way, all the centers of these two circles lie on their radical axis. What is said? Look carefully what is said.
The centers of all the circles which intersect the two given circles orthogonally lie on their radical axis. Where do we lie? Near their radical axis.
We lie on their radical axis. That means, if a circle is cutting them orthogonally, if it is cutting them orthogonally, then the circle that will be there, the center of that circle, that will always lie on the radical axis. That will lie on the radical axis.
The center of the circle which intersects two given circles will lie on the radical axis. The centre of that will lie on the radical axis. So, we have also proved this in the class.
Why will it be like this? We have proved this also. Okay?
So, here the proof cannot be done completely. But, this can be told. It can be reminded to you. That yes, whichever circle is cutting these two circles orthogonally, then that circle will be that circle.
The center of the circle will always lie on the radical axis. If we make a tangent here, it will pass through its center. And if we make a tangent here, it will pass through its center.
Something like this. So, centers of all the circles, centers of the circle which intersect, two given circles orthogonally center of circles which intersect which intersect which intersect two given circles to given circles orthogonally lie on their radical axis. They lie on their radical axis. Now, here is an important point.
What is the use of the Radical Center? Suppose we have three circles and we want a circle that cuts those three circles in an orthogonal manner. There are three circles and we need the circle which intersects them in an orthogonal manner.
So what do we do? We find the radical axis of the three circles. We find the radical axis of the three circles.
And this radical axis which we have found, suppose this point is radical center. So if we take radical center as center and along with that, we draw a tangent from any of the circles. We draw a tangent from this point C. to any one of the circles. And we will draw a circle with its radius where it touched this.
Then this circle will intersect the three given circles orthogonally. So if I want a circle which cuts all three circles orthogonally, then its center will be its center will be radical center. The center of such a circle will be the radical center.
The centre of that will be the radical centre of the circle. Let's make the diagram a little bit better. This will be something like this. The circle which is cutting all three circles orthogonally, that circle will be nothing but... Will be nothing but it will be having the radical center as the center of, as its center.
And what will be the radius? It will be equal to the length of the tangent made on any circle. So circle intersecting, circle intersecting, three given circles, three given circles. Orthogonally. has center at radical center, has center at radical center.
Circle intersecting three given circles orthogonally has center at radical center and radius equal to length of tangent from, length of tangent from, Radical center from radical center to any one of the three given circles. Any one of the three given circles. The length of the tangent on any one of the three circles will be its radius. So here at last we have kept two important things in front of you.
First thing is, if there are two given circles, then the circle which will cut both of them orthogonally, whose center will be applied? Radical axis will be applied. And if I want a circle which will cut three circles orthogonally, then the center of those three circles will be its center. And what will be its radius? What will be its radius?
From this point, what will you make on any circle? Make it tangent. Before going, I want to ask you a question. Please answer me.
In radical axis, we had told you that radical axis is a point locus whose power is equal to two circles. i.e. PA1 x PB1 is equal to PA2 x PB2. Such point locus is called radical axis.
Can I say this too? It is defined as the locus of a point from which the tangents drawn to the two circles are equal. Two circles are equal.
Can I say this or not? Think and tell me. If the tangents drawn to the two circles are equal, then the locus of that point will be the radical axis.
Radical axis is called. Is this right to say? This will not happen.
Do tell me in the comment box below. Okay. If we can say this, then when can we say this?
In every case, can we say this? That the length of the tangent created from every point of the radical axis is always equal on both the circles. That means, if I take a point on the radical axis, I made a tangent on this, I made a tangent on this.
Will the length of both always be equal? Will it be or not? If it will be, then when will it be?
That is where the point should be. Think a little and tell me. Because the correct definition is that what is the radical axis? The power of such points should be equal to the power of both the circles. But if the power is equal, suppose we have the power of P in this respect and the power of S1 in this respect, then the length of tangent is √S1.
So √S1 is for this and √S1 is for this. √S1 is for this circle and √S1 is for this. So if S1 is equal to both, then root S1 will be equal to length of tangent will be equal to that.
So can't we define radical axis like this? So this is the answer I want in the comment box below. Because this will help us to know that you have covered this chapter at what level you have prepared it.
So that's it for today. And that is all from my side as of today. Let's meet in the next Mind Map.
Till then, Jai Shri Krishna. Radhe Radhe.