Let Tbe the line passing through the points P(−2,7)and Q(2,−5). Let F1be the set of all pairs of circles (S1,S2)such that Tis tangent to S1at Pand tangent to S2at Q, and also such that S1and S2touch each other at a point, say, M. Let E1be the set representing the locus of Mas the pair (S1,S2)varies in F1. Let the set of all straight lines segments joining a pair of distinct points of E1and passing through the point R(1,1)be F2. Let E2be the set of the mid-points of the line segments in the set F2. Then, which of the following statement(s) is (are) TRUE?The point (−2,7)lies in E1(b) The point (54,57)does NOT lie in E2(c) The point (21,1)lies in E2(d) The point (0,23)does NOT lie in E1

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Let Tbe the line passing through the points P(−2,7)and Q(2,−5). Let F1​be the set of all pairs of circles (S1​,S2​)such that Tis tangent to S1​at Pand tangent to S2​at Q, and also such that S1​and S2​touch each other at a point, say, M. Let E1​be the set representing the locus of Mas the pair (S1​,S2​)varies in F1​. Let the set of all straight lines segments joining a pair of distinct points of E1​and passing through the point R(1,1)be F2​. Let E2​be the set of the mid-points of the line segments in the set F2​. Then, which of the following statement(s) is (are) TRUE?The point (−2,7)lies in E1​(b) The point (54​,57​)does NOT lie in E2​(c) The point (21​,1)lies in E2​(d) The point (0,23​)does NOT lie in E1​

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Question Text	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 lines 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 $(- 2, 7)$ 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 $E_{1}$
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