 So, let's take some questions based on this, first question, find the angle between, find the angle of intersection, find the angle of intersection of the circles S given as x square plus y square minus 4x plus 6y plus 11 equal to 0 and S dash which is x square plus y square minus 2x plus 8y plus 13 equal to 0, find the angle of intersection between them, first I will just give you 30 seconds, 30 seconds, please type it in the chat box, what is the angle that you get, yes guys tell me the answer, let's say the center of S is C1, so what is C1? C1 will be 2, minus 3, C2 will be, C2 will be 1, minus 4, okay, radius of this will be under root of 4 plus 9 minus 11 which is under root 2, radius of the second circle will be 1 plus 16 minus 13 which is equal to 2, so use the formula cos of theta is D square, now what is the D? D is the distance between them, so D will be under root of 1 square plus 1 square which is root 2, so cos of theta is going to be, cos of theta is going to be D square which is 2 minus 4 plus 2 by 2 into 2 into root 2, so that becomes minus 4 by 4 root 2 which is minus of 1 by root 2, so theta is going to be 135 degree, next question, find the equation of the circle, find the equation of the circle, find the equation of the circle which cuts, which cuts x square plus y square plus 5x plus 7y minus 4 equal to 0 orthogonally, orthogonally and has its center on the line x equal to 2 and passes through and passes through 4, minus 1 and passes through 4, minus 1, anybody any answer please feel free to type on the chat box, so we will say let the desired circle be, let the circle be x square plus y square plus 2 gx plus 2 fy plus c equal to 0 and this circle is orthogonal to the given circle x square plus y square plus 5x plus 7y minus 4 equal to 0, so remember the condition for orthogonality, 2g1g2 plus 2f1f2 is equal to c1 plus c2, so I will write 2g and what is g2, so for this given circle g2 is going to be 5 by 2, similarly 2f1f2, so 2f2 will be 7 by 2, is equal to c1 plus c2, c1 plus c2 is c minus 4, so first condition that you get is 5g plus 7f is equal to c minus 4, this is your first condition, next is the fact that the center lies on this line x equal to 2, so center is minus g minus f, so minus g minus f lies on x equal to 2, so that clearly implies minus g is equal to 2, so g is equal to minus 2, so one achievement and the circle also passes through 4 comma 1, so 4 comma minus 1 will satisfy this equation, so you will get 16 plus 1 plus 8g minus 2f plus c equal to 0, so you get 17 plus 8g minus 2f plus c equal to 0 and g is only minus 2, so you can only put it over here, so this is going to be minus 16 minus 2f plus c equal to 0, that means 1 minus 2f plus c is equal to 0 and from here I will get minus 10 plus 7f minus c plus 4 equal to 0, add them, when you add them you would get minus 9 plus 5f plus 4 equal to 0, so f becomes 1, f becomes 1 and c would become 2f minus 1 which is also 1, so we have got all the required parameters, so equation will be x square plus y square minus 2gx, sorry plus 2gx will be minus 4x, 2f y will be 2y plus 1 equal to 0, this would become your answer. Guys it is clear, now very important information for you, equation of a circle, equation of a circle cutting three circles orthogonally, cutting three circles orthogonally, this is given by the determinant, it is given by the determinant x square plus y square xy1 minus c1 g1 f1 minus 1 minus c2 g2 f2 minus 1 and minus c3 g3 f3 minus 1 determinant equal to 0, equal to 0. So guys the proof for this you will provide me on the group, so how come this becomes the equation of a circle cutting three circles, so I am assuming that the three circles are having the equation 2gix 2fi y plus ci equal to 0 where i can take values from 1 to 3, so you prove this and send me the solution on the group. Meanwhile based on this we can directly take up a question, please note this down, so later on I will tell you a easier way to get to this equation to the concept of radical center and radical axis, so meanwhile let us take a question, find the equation of the circle, find the equation of the circle which cuts the following orthogonally, which cuts the following circles orthogonally, so s1, s2 or you can say s dash, s double dash, s triple dash, so equations are as follows, x square plus y square minus 2x plus 3y minus 7 equal to 0, x square plus y square plus 5x minus 5y plus 9 equal to 0, x square plus y square plus 7x minus 9x plus 29 equal to 0, guys that determinant method is one of the methods, you do not have to follow it if you are not comfortable expanding a 4 by 4 determinant, you can also take the approach of 2g1g2 plus 2f1f2 equal to c1 by c2, so three equations you will get, three unknowns you can easily solve for them and if you are trying to follow the determinant approach x square plus y square xy1 minus c1 will be 7 minus c2 will be minus 9, this will be minus 29, g1 will be minus 1, g2 will be 5 by 2, this will be 7 by 2, f1 will be 3 by 2, minus 5 by 2, minus 9 by 2 and this will be minus 1, minus 1, minus 1 equal to 0, so you can do one thing, you can do this operation, r1 is r1 plus r4, r2 as r2 minus r4 and r3 as r3 minus r4, okay, so these operations can be done plus r4 will make this 0, so you get x square plus y square minus 29, x plus 7 by 2, y minus 9 by 2 and you get a 0 over it, okay and when you subtract you get 36, you get minus 9 by 2 and you get 6 and 0, okay, again when you subtract here you get a 20, 20 and you get minus 1, you get 2, you get 0 and again you get minus 29, 7 by 2, minus 9 by 2, minus 1, so you can expand with respect to the fourth column, right, so you can write this as, you can write this as, so what is the position, this is the fourth row fourth column, so minus 1 to the power 4 plus 4 which is going to be one itself times the determinant, so please write down the desired determinant equal to 0, so let us expand this, so minus sign becomes immaterial because anyways we have a 0 on the, on the right side, so this will give you x square plus y square minus 29 times minus 9 plus 6 which is minus 3 and minus x plus 2 times you will have 72 minus 120, 72 minus 120 which is going to be 48 plus 48, okay and finally you will have, and finally you will have y minus 9 by 2 times minus 36 minus 36 plus 90 which is 54 equal to 0, correct, so let us expand this, so when you expand this you can straight away drop a factor of 3 from everywhere, so you have x square plus y square minus 29 minus 16 x plus 7 by 2 and minus 18 y minus 9 by 2 equal to 0, so it will be x square plus y square minus 16 x minus 18 y and the remaining constants will be minus 29 minus 56 plus 81 plus 81 equal to 0 which is x square plus y square minus 16 x minus 81 18 y and I think this will become minus 4 equal to 0 it becomes 84 85 minus 85 plus so this will be your answer, is that clear guys, you could have also assumed the equation to be x square plus y square plus 2 g x plus 2 f y plus c equal to 0 and individually with all the circles you could have written the condition for orthogonality, three equations, three unknowns you can be, you can easily solve them, any questions, please feel free to type in if there is any questions.