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 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?

• A. The point $(-2, 7)$ lies in $E_1$
• B. The point $\left( \frac{4}{5} , \frac{7}{5} \right)$ does NOT lie in $E_2$
• C. The point $\left( \frac{1}{2} , 1 \right)$ does NOT lie in $E_2$
• D. The point $\left(0 , \frac{3}{2} \right)$ does NOT lie in $E_1$

Answer 1

Answer: (A), (D)

$\angle PMQ = 90^{\circ} $
$\Rightarrow \frac{\beta+5}{\alpha-2} \times \frac{\beta-7}{\alpha+2} = - 1 $
Locus of M
$x^{2} + y^{2} -2y -39=0$ ....(1)
Equation of chord of circle whose midpoint is $(\alpha, \beta) $ is
$ S_{1} = T $
$\Rightarrow x\alpha+y\beta-\left(y +\beta\right) - 39 = \alpha^{2} + \beta^{2}+2\beta-39$
$ \Rightarrow \alpha^{2} + \beta^{2} -2\beta -\alpha+1=0$
Locus : $ x^{2} + y^{2} - x - 2y + 1 = 0 $
Option (A) is incorrect although it satisfies eq. (1) otherwise the line T would touch the second circle on two points. (4/5, 7/5) satisfies eq. (2) while (1/2, 1) does’t. (0, 3/2) does not satisfy (1).