Let T be the line passing through the points $P (- 2, 7) a n d Q (2, - 5) .$ 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 $S_{1} & S_{2}$ touch each other at 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?

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

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

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

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

answer is B, D.

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Detailed Solution

MN is radical axis and MN = NP = NQ
locus of M is circle with centre N = $N = \frac{P + Q}{2}$ and $r = N Q o r N P = \sqrt{40}$
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Locus of M is circle centre (0,1) and radius $\sqrt{40}$ , excluding points P and Q
$\Rightarrow x^{2} + y^{2} - 2 y = 39$ excluding points P and Q
$∴ (- 2, 7)$ is excluded from locus and $(0, \frac{3}{2})$ does not lie in $E_{1}$
Locus of point in $E_{2}$ is circle with diameter (0,1) and (1,1), excluding midpoints of chords passing through P or Q
$\Rightarrow x^{2} - x + {(y - 1)}^{2} = 0$
$∴ (\frac{1}{2}, 1)$ does not lie in $E_{2}$ and $(\frac{4}{5}, \frac{7}{5})$ doesn’t lie in $E_{2}$ since latter is midpoint of chord through P