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).