hello everyone welcome to senior pablo tv today we will be discussing the circle we are pre-calculus first lesson or the mo in your module one the circle so first let us define what is a circle a circle is the set of all points in a plane that are at a constant distance the reduce from the fixed point the fixed point is called the center of the circuit so let us illustrate to better understand so we have a circle and we have a fixed point that is the center so from the center we have a set of points okay this is infinite many points and these points are equidistant from our center that means we were going to connect one point on the circle to our center that is our reduce again all the points around the circle is equidistance from the center and if we're going to connect a point to another point that passes through the center that is what we call the diameter and our measurement radius is half of a diameter the ammeter is twice the radius so that is the circle in our circle we have the standard form this is the standard form quantity x minus h squared plus quantity y minus k squared is equal to r squared we're in r is our reduce and if we're going to get the value of x or y we will find the center the center is the h k let's say you have x minus 3 squared plus x minus 4 squared is equal to 3 squared equate x minus 3 to 0 so your h will be positive 3 and your k x minus 4 equals 0 k will be positive four so let us discuss one by one later to better understand next we have the general form x squared plus y squared plus dx plus e y plus f is equal to zero and in our general form we have a shortcut formula for the center negativity over 2 and negative e over 2 that is the shortcut formula for the center okay to better understand the center let us first discuss transforming general form into standard form let's have the following examples and now let's have this example transform the general form x squared plus y squared minus 10x plus 4y minus 7 is equal to 0 of a circle into standard form find the center and radius so from general form we're going to transform into standard form so in our introduction we have this general form going to our standard form then after getting the standard form we need to find the radius and the center we can transform by using the completing the square your lesson when you were in grade 9 so let's try to solve this problem so first step rearrange our given okay the given is x squared plus y squared minus 10 x plus 4y minus 7 is equal to zero this is our given rearrange so we have x squared minus 10 x so x and x must be written besides each other then plus y squared plus four y is equal to addition property so in short we're going to transpose that will become positive seven now regroup make the expression a perfect square trinomial so we have x squared minus 10x plus blank we're going to add the number to make it a perfect square trinomial plus y squared plus four y plus blank plus blank is equal to positive seven okay we will add a block on the left side of the equation so we need to add blank here and another block to make the equation balance now make our expression inside the parentheses a perfect square trinomial get the middle term the middle term is negative 10 divided by 2 so 10 divided by 2 that is 5 so negative 5 then 5 squared 5 squared 25 so we need to add 25 here also add 25 on the right side of the equation next 4 divided by 2 that is 2 2 square that is 2 times 2 which is 4. so we added 4 on the left side add 4 on the right side of the equation okay now a perfect square trinomial and a perfect square trinomial we're going to write it into square of a binomial so square of a binomial so that is x copy the sign of the middle term minus square root of 25 that is five then right square plus y sine of the middle term plus square root of four positive 2 squared is equal to so we need to add 7 plus 25 that is 32 plus 4 we have 30. so this is now our standard form this is now our standard form if you want to master a perfect square trinomial go to our grade 9 playlist to recall your completing this square now find the center and reduce so the center we have the hk so let's get x minus five equated to zero so our x is equal to positive five and y plus two equated to zero so y is equal to negative two now the center is so our center is equal to our x or h is positive 5 and our k negative that is now the center and the radius just get the square root of our constant term the square root of 36 square root of 36 that is positive six so maybe you will say sir positive negative six because that is because 36 is a perfect square number so since we are dealing with a distance or a measurement we're just going to get the positive volume so the radius is positive 6. so that's on how to transform form into standard form now your turn i want you to answer this problem the problem is transform the general form x squared plus y squared plus 2x minus 6y minus 15 equals 0 of a circle into standard form find the center and the radius pause the video and answer this problem then after answering resume watching to check your answers okay here's the solution copy the given general form x squared plus y squared plus 2x minus 6y minus 15 is equal to zero now rearrange our given so x squared plus 2x plus y squared minus 6 y is equal to after rearranging move the constant on the right side or add positive 15 to cancel negative 15 let's see next make our array group and make it a perfect square trinomial group x squared plus 2x plus 1 plus blank plus y squared minus 6y plus blood is equal to 15 we added two blocks to make the equation balance and now make the expression a perfect square trinomial middle term in this case we have 2 so 2 divided by 2 that is one one square one times one positive one get the middle term that is negative six so negative six divided by two that is negative three negative three times negative three positive nine we added one and we added nine now make our perfect square trinomial a square of a binomial x sine of the middle term plus square root of one positive one then don't forget to write square square of a binomial plus y sine of the middle term minus square root of 9 3 squared is equal to 15 plus 1 plus 9 that is 25 this is now our standard form we are now ready to find the center and the regions so the center x plus one equates to zero and y minus three equate to zero so x is equal to negative one and y is equal to positive three so the center now is negative one positive and the radius that is the square root of our constant is 25 is 25 a perfect square number yes so square root of 25 is positive 5 so the radius is positive 5. so that's on how to transform general form into standard form for our next video we're going to discuss transforming a standard form into general form in our previous discussion we discuss transforming general form into standard form in this video we're going to discuss standard form going to or transforming to general form if you haven't watched the video i will put it in our cart just click the cart then it will direct you in that video let us recall the standard form is x minus h of quantity x minus h squared plus quantity y minus k squared is equal to r squared and the general form x squared plus y squared plus d x plus z y equals f is equal to zero our lesson transforming standard form to general form let's have this problem number one find the general form of the circle quantity x plus five raised to two plus quantity y minus three twisted two is equal to forty-nine we're going to transform into this form okay let's start our first step we need to expand our square of a binomial gradient lesson special products first step is square the first term square that will become x squared second step multiply the first and the second term so x times five that is five x times two that is ten x again multiply the first and the second term times two that's why we have ten x plus square the second term five square give us twenty five copy the plus sign now square the first term that is y squared multiply the first and the second term that is negative 3y times 2 negative 6y and square the second term positive 9 is equal to 49. just copy 14. and now combine like terms so x squared plus 10 x we can combine 25 and nine here so let's just copy first plus y squared minus six one 25 plus nine that will give us 34 is equal to 49 and now let us rearrange our equation so x squared copy positive y squared for our 10 plus 10x minus 6y plus 34. in our general form that is equated to zero so we need to subtract 49 is equal to zero let us combine x squared plus y squared plus 10x minus 6y 34 minus 49 that is negative 15 is equal to zero this will be the general form of the circle if your teacher asks you find the value of d e f so the value of d is then value of e is e negative six value of x is negative if your teacher asks you to find the value of v e and f okay that on how to transform standard form to general form now your turn i want you to answer the next problem your problem is find the general form of the circle quantity x plus 5 raised to 2 plus quantity y minus 1 raised to 2 is equal to 9. find the value of d e and f okay let's find the general general form so copy x plus 5 raised to 2 plus 1 minus a y minus 1 raised to 2 is equal to 9. a square of a binomial so let us expand square the first term that is x squared multiply the first and the second times two is positive ten x and square the second term five square positive twenty five plus square the first term y squared multiply the first and the second term times two negative two one square the second term positive one is equal to positive nine now combine like terms while combining like terms you can now rearrange our equation x squared let us arrange now plus y squared plus 10 x minus 2y so 25 plus 1 that is positive 26 is equal to 9. again our general form is equated to zero so great x squared plus y squared plus 10x minus 2y plus 26 minus 9 is equal to 0. let's combine 26 minus 9. so x squared plus y squared plus 10 x minus 2y 26 minus 9 that is opacity of 15 is equal to zero this is now our standard form find the value of d e and f so our d is equal to d then e is negative 2 and f is that is transforming standard form to general hello everyone welcome to senior pablo tv so after discussing transforming standard form to general form and vice versa now let's going to discuss what if the given are the center and the regions we're going to find the standard form or the general form let's have our problem number one find the equation of a circle in standard form whose center is at negative 4 3 and radius is square root of 5. so center is negative 4 positive 3 and the radius is the square root of 5. the standard form of a circle is x minus h raised to two plus quantity y minus k raised to two is equal to r squared this is the standard form we're going to write into this form so the given is the center and the radius let's just go into substitute the center center focus on our center so the standard form is quantity x minus h so we have x just reverse the sign so negative 4 our k will become 5 that will become positive 4 raised to 2. then plus y reverse the sign will become the plastic will become negative three raised to two is equal to our r is square root of 5. we're going to square so our final answer is x plus four raised to two plus y minus three this two is equal to we were going to simplify this square root and the square will be cancelled out is equal to 5. so this will be the standard form of our circle if the given is the center and the radius okay that's it maybe sir why did you change the sign that is negative 4 will become positive 4 because if we're going to solve this step by step that is x is equal to negative 4. so our x is equal to negative 4. move on the left side of the equation that will become x minus four is equal to zero this will be the number inside our parenthesis positive four that's why we have x plus 4 and so y positive 3 will become negative 3 because we have x sorry that is y y is equal to 3 of the left side y minus three y minus three y minus three now i was going to find out the relationship between the center the radius and the standard form or the equation of the circle complete the table below so we're here we're going to find the center and the radius next will be the equation of the circle now i will going to answer one and two then you're going to answer three four five and six okay so the center center here x is equal to positive four so that will become positive four y is positive one just change the sign our radius get the square root of 25 square root of 35 so our radius is five that would be the answer next given the center and the ranges we're going to find the equation of the circle so x change the sign plus seven negative two plus y minus three raised to two is equal to eight square this time we're going to square eight times eight 64. okay now try the fourth fifth third fourth fifth and sixth even you can pause the video and after answering resume watching to check your answers okay the answers are center is zero zero so that will become x plus zero raised to two but since this is zero we're just going to write x squared plus y squared is equal to we're going to square so positive six just remove this square root next right in the center change the sign negative three negative one and the radius get the square root square and of seven next x minus four raised to two plus y change the sign plus to 2 is equal to square positive and our last notice that this is x squared that's not written in our parentheses so the center here for our x that will become zero zero and for the y change the sign positive one again you don't have parentheses here that means the value of x is zero change the sign and now to get the radius get the square root the square root of 36 is positive and we're done that's on how to find the equation of the circle in standard form giving the radius and the center of the circle thank you for watching senor pablo tv and for our last video for the circle we're now going to graph the center our now going to graph the circle given the center the radius or the standard form of the circle this will be our fourth and last video in our topic the circle in this video we will be or you will learn on how to graph the circle let us first familiarize our cartesian plane so we have the x-axis the horizontal line and the vertical line which is the y-axis and on the right side that is the positive numbers up positive numbers on the left side the negative numbers and below are the negative numbers of course this is what we call the origin zero zero zero is the coordinates of the origin now the problem is the general form of the circle is 4x squared plus 4y squared plus 8x minus 16y minus 80 is equal to 0. find the center radius and standard form okay let us continue off the circle and graph and graph the circle let us find first our standard form so copy the given the given is 4 x squared plus 4 y squared plus z x minus 16y minus 80 is equal to zero analyze our given the equation is divisible by four so we can reduce our given equation so divisible by four so divide by four this will become x or divided by four will become 1 x squared versus the x squared plus 4 divided by 4 becomes y squared plus 8 divided by 4 becomes 2x minus 16 divided by 4 that is 4 copy y minus 80 divided by 4 20 is equal to zero this is now the reduced form of our genome form now we arrange our general form that will become x squared plus 2x plus y squared minus 4y is equal to positive 20. make our expression a pst perfect squared trinomial so we add love plus y squared minus 4y plus blood is equal to 20. we added two blocks so plus blog and plus make it a perfect square trinomial middle term is two so two divided by two that is one one square positive one x negative four divided by two that is two three square positive four we added one so add one adding four and four now write it into sphere of a binomial so x sine of the middle term positive square root of one is one square plus y minus square root of 4 is 2 squared is equal to add 25 this is our standard pool standard form let us find the center and the radius the center is 1 change the sign so negative one and positivity and our reduce the square root of 25 that is positive five this will be our ranges now let us graph in our cartesian plane to graph first plot our center our center is negative one positive two where is negative one negative one for x and positive two four y so this is our center our radius is five to find the other points let us count from the center five units up so one two three four five that will be our topmost point rightmost count five units from the center one two three four five okay again one two three four five that would be four positive two and the left most from the center count five one two three four five okay negative six and bottom lows count five below one two three four five now connect the circle or connect the points of the circle so it should be circled assume that this is a perfect circle okay this is now our circle so that's on how to graph the circle in our cartesian plane so this ends our lesson for your assignment given the general form of 4 x squared plus 4 y squared plus 12 x minus 4y minus 90 is equal to zero find the center radius and the standard form of the circle and graph in your cartesian plane assume that this is a perfect circuit it's hard to graph here because it's quite slippery thank you for watching senior pablo tv and we're done in your module one the circle kindly invite your classmate and share this video to your classmates so that they can answer and enjoy answering the questions in your module thank you so much this is senor pablo see you in your module 2 the parabola [Applause] you