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…

Question

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 , &#160; S 2 varies in F 1 . Let the set of all straight line segments joining a pair of distinct points of E 1 and passing through the point R 1,1 be F 2 . Let E 2 be the set of the mid-points of the line segments in the set F 2 . Then, which of the following statement(s) is (are) TRUE?

Aicharya · Accepted Answer

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 of E 2 is x 2 + y 2 - x - 2 y + 1 = 0 But if P is not the part of locus of M then P Q is not the chord of locus of M . So point 4 5 , 7 5 does not lie in E 2 . 1 2 , 1 does not satisfy the equation hence does not lie in E 2 .

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…

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