class 11

Math

Co-ordinate Geometry

Conic Sections

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 $(54 ,57 )$does NOT lie in $E_{2}$(c) The point $(21 ,1)$lies in $E_{2}$(d) The point $(0,23 )$does NOT lie in $E_{1}$