If a rectangular hyperbola (x – 1)(y – 2) = 4 cuts a circle x^2 + y^2 + 2gx + 2fy +c = 0 at points

Q: If a rectangular hyperbola (x – 1)(y – 2) = 4 cuts a circle x² + y² + 2gx + 2fy +c = 0 at points (3, 4), (5, 3), (2, 6) and (-1, 0), then the value of (g + f) is equal to

(A) 8

(B) -9

(C) -8

(D) 9

Sol. Let x_iy_i, i = 1, 2, 3, 4 be the points of intersection of hyperbola and circle, then

$\Large \frac{\Sigma x_i}{4} = \frac{-g+1}{2} $

g = -7/2

and $\Large \frac{\Sigma y_i}{4} = \frac{-f+1}{2} $

∴ f = -9/2

g + f = -8.

Hence (C) is the correct answer.