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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 segment joining a pair of disntinct point 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? |
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Answer» The point (-2, 7) lies in `E_(1)`  `THEREFORE MN=NP=NQ` `therefore` Locus ofM ISA CIRCLE having PQ as its diameter of circle. `therefore` Equation of circle `(X-2) (x+2)+(y+5)(y-7)=0` `rArrx^(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. |
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