If a rectangular hyperbola $(x - 1) (y - 2) = 4$ cuts a circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ at points $(3, 4), (5, 3), (2, 6)$ and $(- 1, 0)$ , then value of $g + f + c$ is equal to :

- 6

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Solution

The correct option is A $- 6$
Substituting $(3, 4)$ and $(- 1, 0)$ in the equation of circle as they lie on it, we get
$9 + 16 + 6 g + 8 f + c = 0$ ....(1)
$1 + 0 - 2 g + c = 0$ ...(2)

Subtracting (2) from (1), we get
$\Rightarrow 24 + 8 g + 8 f = 0$
$\Rightarrow g + f + 3 = 0$
$\Rightarrow g + f = - 3$ ... (3)

Substituting $(5, 3)$ and $(2, 6)$ in the equation of circle as they also lie on it, we get
$4 + 36 + 4 g + 12 f + c = 0$ ...(4)
$25 + 9 + 10 g + 6 f + c = 0$ ...(5)
From (3), (4) and (5), we get
$g = - 1$ , $f = - 2$
$c = - 3$
$g + f + c = - 1 - 2 - 3 = - 6$

Hence, option A.

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