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 tangent 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 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?

The point (4/5, 7/5 ) does NOT lie in E 2

The point (0, 3/2 ) does NOT lie in E 1

The point (1/2, 1 ) lies in E 2

Mathematics · Circles

JEE Advanced 2018 — Paper 2 — Question 29

Let T be the line passing through the points $\mathrm{P}(-2,7)$ and $\mathrm{Q}(2,-5)$ . Let $\mathrm{F}_{1}$ be the set of all pairs of circles $\left(S_{1}, S_{2}\right)$ such that $T$ is tangent 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 $\left(S_{1}, S_{2}\right)$ 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?

Option A:
The point $(-2,7)$ lies in $\mathrm{E}_{1}$
Option B:
The point $\left(\frac{4}{5}, \frac{7}{5}\right)$ does NOT lie in $\mathrm{E}_{2}$
Correct
Option C:
The point $\left(\frac{1}{2}, 1\right)$ lies in $\mathrm{E}_{2}$
Option D:
The point $\left(0, \frac{3}{2}\right)$ does NOT lie in $\mathrm{E}_{1}$
Correct

Answer: B, D

Step-by-step solution

Locus of M is circle center $(0,1)$ and radius $\sqrt{40}$ , excluding points P and Q

$\Rightarrow x^{2}+y^{2}-2 y=39$ excluding points $P$ and $Q$

$\therefore \quad(-2,7)$ is excluded from locus and $\left(0, \frac{3}{2}\right)$ does not lie in $\mathrm{E}_{1}$

Locus of point in $E_{2}$ is circle with diameter $(0,1)$ and $(1,1)$ , excluding midpoints of chords passing through

P or Q

$\Rightarrow \mathrm{x}^{2}-\mathrm{x}+(\mathrm{y}-1)^{2}=0$

$\therefore\left(\frac{1}{2}, 1\right)$ does not lie in $\mathrm{E}_{2}$ and $\left(\frac{4}{5}, \frac{7}{5}\right)$ doesn't lie in $\mathrm{E}_{2}$ since latter is midpoint of chord through P

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2018
Paper: Paper 2
Subject: Mathematics
Chapter: Circles
Topic: Tangent & Normal , pair of tangents to circle , chord of contact