 Hello and welcome to the session. In this session we will discuss a question which says that proof that the circles x square plus y square minus 8x plus 6y plus 16 is equal to 0 and x square plus y square is equal to 0 touch externally at a point also find its point of contact. Now before starting the solution of this question we should know some results. First is circles between the centres of the two circles the radii of the two circles for internal contact is equal to this circle are given by my circle is given by us with the solution. Now the equation of the first circle is given to our equation of the circle. Now comparing this with the journal equation of the circle of the circle which is given by that is the equation number 1 is minus 2 minus f of g and f here this will be equal to minus of minus 4 first circle which is given by equation number 1 is of g square minus c. Now c is 6 is g and this is f this will be equal to square root of minus 4 square plus 3 square minus 16 which is equal to square root of 16 plus 9 minus 16 which is equal to root 9 which is equal to now given the equation of the second circle minus 20 is equal to 0. Now comparing this with the journal equation of the circle f is equal to minus 10 which implies g is equal to 2 equation number 2 which is given by equation number 2 is minus 2 minus f this is g and this is f here so this is equal to minus which is equal to minus 2 5. Now r2 of the circle which is given by equation number 2 which is minus 5 square minus minus 20 so it will be minus of minus 20 25 which is equal to 7 therefore r2 is equal to the coordinates of center c1 are minus 3 and the coordinates of center c2 so by distance c1 c2 is equal to square root of x1 whole square equal to square root of minus 2 minus 4 whole square plus n is 3 whole square which is further equal to square root of minus 4 is minus 6 and minus 6 square is 2 square root of 100 which is equal to so the distance c1 c2 is equal to 10 units currently at a point. Now using the first result which is given in the key idea is equal to 3 which is equal to 10 units in the centers which is c1 c2 is equal to the point of contact of the two circles so the second result which is given in the key idea now the point of contact let it be p xy divides the line joining the centers of the two circles that is 1 is 3 and r2 is 7 units so the ratio will be equal to 3 is to 7. Now the point of the two circles c1 c2 find the coordinates of p and that we can find by using the section formula or let this be x1 divided by 1 now by section formula x1 whole square divided by 2 plus m2 divided by 1 now putting all these values here equal to 3 into minus 2 plus and here it will be 3 into 5 plus into minus 3 whole upon 3 upon 10 and here 15 minus 21 whole upon this will be equal to the equation of the given question and that's all for this session hope you all have enjoyed the session.