Let T be the line passing through the points $\mathrm{P}(-2,7)$ and $\mathrm{Q}(2,-5)$. Let $\mathrm{F}_1$ be the set of all pairs of circles ( $\mathrm{S}_1, \mathrm{~S}_2$ ) such that T is tangent to $\mathrm{S}_1$ at P and tangent to $\mathrm{S}_2$ at Q , and also such that $\mathrm{S}_1$ and $\mathrm{S}_2$ 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 $\mathrm{F}_2$. Let $\mathrm{E}_2$ be the set of the mid-points of the line segments in the set $\mathrm{F}_2$. Then, which of the following statement(s) is (are) TRUE ?