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 tangent 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 statement(s) is (are) TRUE ?

(A) The point (-2, 7) lies in E₁  

(B) The point (4/5,7/5) does NOT lie in E₂  

(C) The point (1/2,1) lies in E₂

(D) The point (0,3/2) does NOT lie in E₁

Answer 1

Answer: D

Solution:

Locus of M is circle center (0, 1) and radius √40 , excluding points P and Q
=>x² + y² – 2y = 39 excluding points P and Q
So, (–2, 7) is excluded from locus and (0, 3/2) does not lie in E₁
Locus ofpoint in E₂ is circle with diameter (0, 1) and (1, 1)
=>x² – x + (y – 1)² = 0
So, (1/2,1) does not lie in E₂ and (4/5,7/5) lies in E₂