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 $\left( \frac { 1 } { 2 } , 1 \right)$ lies in $E _ { 2 }$
(D) The point $\left( 0 , \frac { 3 } { 2 } \right)$ does NOT lie in $E _ { 1 }$