Let \( \mathrm{T} \) 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 \( \left(S_{1}, S_{2}\right) \)
\( \mathrm{P} \) such that \( T \) is tangent to \( S_{1} \) at \( P \) and tangent to \( S_{2} \) at \( Q \),
W 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 \( \left(S_{1}, S_{2}\right) \) 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?
(1) The point \( (-2,7) \) lies in \( E_{1} \)
(2) The point \( (4 / 5,7 / 5) \) does NOT lie in \( E_{2} \)
(3) The point \( (1 / 2,1) \) lies in \( E_{2} \)
(4) The point \( (0,3 / 2) \) does NOT lie in \( E_{1} \)
(JEE Advanced 2018)
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