Let T be the line passing through the points P - 2 , 7 and Q 2 , - 5 . Let F 1 be the set of all pairs of circles S 1 , S 2 such that T is tangents to S 1 at P and tangent to S 2 at Q , and also such that S 1 and S 2 touch each other at a point, say, M . Let E 1 be the set representing the locus of M as the pair S 1 , S 2 varies in F 1 . Let the set of all straight line segments joining a pair of distinct points of E

Step-by-step solution

A P = A Q = A M So ∠ P M Q = 90 ° Locus of M is a circle having P Q as its diameter Hence, E 1 : x 2 + y 2 - 2 y - 39 = 0 , x ≠ ± 2 Clearly - 2 , 7 is not included in the locus because we can not consider point circles, and 0 , 3 2 does not lie on E 1 as it does not satisfy the equation above. The equation of chord with B h , k as mid point will be given by T = S 1 or x h + y k - y - k - 39 = h 2 + k 2 - 2 k - 39 or x h + y k - y + k = h 2 + k 2 It passes through R 1 , 1 . So, h + k - 1 + k = h 2 + k 2 or the locus