Let C be a circle passing through the points A 2 , - 1 and B 3 , 4 . The line segment A B is not a diameter of C . If r is the radius of C and its centre lies on the circle x - 5 2 + y - 1 2 = 13 2 , then r 2 is equal to

Step-by-step solution

Equation of line joining A B is y + 1 = 5 x - 2 i.e. 5 x - y - 11 = 0 Also A B = 3 - 2 2 + 4 + 1 2 = 26 r is the radius of the circle C For the circle x - 5 2 + y - 1 2 = 13 2 centre is 5 , 1 and radius R = 13 2 Equation of perpendicular bisector of A B will be y - 1 = - 1 5 x - 5 ⇒ x + 5 y - 10 = 0 Solving this equation of line with x - 5 2 + y - 1 2 = 13 2 , we get M as 5 2 , 3 2 So M lies on the line joining A B . Now r 2 = C M 2 + A M 2 = 2 × 13 2 2 + 13 2 2 r 2 = 65 2