Let the circle $(x − 1) 2 + (y − 2) 2 = 25$ intersects rectangular hyperbola with transverse axis along $y = x at 4$ points A,B,C and D having coordinates $x i, y i, i = 1, 2, 3, 4$ such that origin is centre of hyperbola. If $l = x 1 + x 2 + x 3 + x 4, m = x 12 + x 22 + x 32 + x 42 and$ $n = y 12 + y 22 + y 32 + y 42, Then m + n l =$

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answer is 50.

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Detailed Solution

Let hyperbola is $x y = C 2$ then $c t 1 c t$ lies on circle.
$∴ c 2 t 4 − 2 c t 3 − 20 t 2 − 4 c t + c 2 = 0$
Then $∑ t 1 = 2 c, ∑ t 1 t 2 = − 20 c 2, ∑ t 1 t 2 t 3 = 4 c, t 1 t 2 t 3 t 4 = 1$ $l = 2, m = 44, n = 56, m + n l = 1002 = 50$

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