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 `(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`

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)` lie in `E_(2)`
D. The point `(0,3//2)` does NOT lie in `E_(1)`

Answer

2,4

Explanation

According to the question , we have the following figure `:`

Clearly, `/_ PMQ =(pi)/(2)`.
Hence, locus of M is
`(x+2)(x-2)+(y-7)(y+5)=0`
or `E _(1) -= x^(2)+y ^(2)-2y-39=0` (i)
Clearly, `E_(2)` is the locus of mid-points of chords of `E_(1)` passing through (1,1).
Let the mid-pioint of chord of `E_(1)` be (h,k) .
Its equation is
`h+k- (1+k) -39= h^(2)+k^(2)-2k-39`
or `h^(2)+k^(2)-2k-h+1=0`
Hence, `E_(2) -= x^(2)+y^(2)-x-2y+1=0` (ii) ltbrlt Operation (A) is incorrect, although, it satisfies eq. (i) otherwise the line T would touch the second circle at two points.
Als, `(4//5,7//5)` satisfies eq. (ii) but again in this case one end of the chord would be `(-2,7)` which is not included in `E_(1)`.Therefore, `(4//5, 7//5)` does not lie in `E_(2)`
`(1/2,1)` does not satisfy eq. (ii), therefore, it does not lie in `E_(2)`.
`(0,3//2)` does not satisfy eq. (i), so it does not lie in` E_(1)`.