JEE Advanced | 2018CircleHard

Question

Let T be the line passing through the points P(–2, 7) and Q(2, –5). Let F1 be the set of all pairs of circles (S1, S2) such that T is tangents to S1 at P and tangent to S2 at Q, and also such that S1 and S2 touch each other at a point, say, M. Let E1 be the set representing the locus of M as the pair (S1, S2) varies in F1. Let the set of all straight line segments joining a pair of distinct points of E1 and passing through the point R(1, 1) be F2. Let E2 be the set of the mid-points of the line segments in the set F2. Then, which of the following  tatement(s) is (are) TRUE ?

Options

A.

The point (–2, 7) lies in E1

B.

The point45,75 does NOT lie in E2

C.

The point12,1 lies in E2
 

D.

The point0,32 does NOT lie in E1

Solution


AP = AQ = AM
Locus of M is a circle having PQ as its diameter
Hence, E1 : (x – 2) (x + 2) + (y – 7)(y + 5) = 0 and x ≠2
Locus of B (midpoint)
is a circle having RC as its diameter
E2 : x(x – 1) + (y – 1)2 = 0
Now, after checking the options, we get (D)

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