Let T be the line passing through the points 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)` 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 ?

Options

A. The point (-2, 7) lies in `E_(1)`
B. The point `((4)/(5),(7)/(5))` does NOT lie in `E_(2)`
C. The point `((1)/(2),1)` lies in `E_(2)`
D. The point `(0,(3)/(2))`does NOT lie in `E_(1)`

Answer

A, D

Explanation

It is given that T is tangents to `S_(1) and S_(2)` at Q and `S_(1) and S_(2)` touch externally at M.

`therefore MN=NP=NQ`
`therefore` Locus of M is a circle having PQ as its diameter of circle.
`therefore` Equation of circle
`(x-2) (x+2)+(y+5)(y-7)=0`
`rArr x^(2)+y^(2)-2y-39=0`
Hence, `E_(1):x^(2)+y^(2)-2y-39=0,xnepm2`
Locus of mid-point of chord (h,k) of the circle E_(1) is
`xh+yk-(y+k)-39=h^(2)+k^(2)-2k-39`
`rArrxh+yk-y-k=h
^(2)+k^(2)-2k`
since, chord is passing through (1, 1). lt brgt `therefore` Locus of mid-point of chord (h,k) is
`h+k-1-k=h^(2)+k^(2)-2k`
` rArr h^(2) +k^(2) - 2k -h +1=0`
Locus is `E_(2): x^(2)+y^(2)-x-2y +1 =0`
Now, after checking options, (a) and (d) are correct.