SATHEE: Circle 2 Question 15

Circle 2 Question 15

16. 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?

(2018 Adv.)

(a) The point $(- 2, 7)$ lies in $E_{1}$

(b) The point $\frac{4}{5}, \frac{7}{5}$ does NOT lie in $E_{2}$

(c) The point $\frac{1}{2}, 1$ lies in $E_{2}$

(d) The point $0, \frac{3}{2}$ does NOT lie in $E_{1}$

Show Answer

Answer:

Correct Answer: 16. (a, d)

Solution:

It is given that $T$ is tangents to $S_{1}$ at $P$ and $S_{2}$ at $Q$ and $S_{1}$ and $S_{2}$ touch externally at $M$ .

$∴ M N = N P = N Q$

$∴$ Locus of $M$ is a circle having $P Q$ as its diameter of circle.

$∴$ Equation of circle

$\begin{array}{cc} (x - 2) (x + 2) + (y + 5) (y - 7) = 0 \\ \Rightarrow & x^{2} + y^{2} - 2 y - 39 = 0 \end{array}$

Hence, $E_{1} : x^{2} + y^{2} - 2 y - 39 = 0, x \neq \pm 2$

Locus of mid-point of chord $(h, k)$ of the circle $E_{1}$ is $x h + y k - (y + k) - 39 = h^{2} + k^{2} - 2 k - 39$

$\Rightarrow x h + y k - y - k = h^{2} + k^{2} - 2 k$

Since, chord is passing through $(1, 1)$ .

$∴$ Locus of mid-point of chord $(h, k)$ is

$\begin{array}{cc} h + k - 1 - k = h^{2} + k^{2} - 2 k \\ \Rightarrow & h^{2} + k^{2} - 2 k - h + 1 = 0 \end{array}$

Locus is $E_{2} : x^{2} + y^{2} - x - 2 y + 1 = 0$

Now, after checking options, (a) and (d) are correct.

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