The condition that the pair of tangents drawn from the origin to the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ . may be at right angles is

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$2 (g^{2} + f^{2}) = c$

$g^{2} + f^{2} = c$

$g^{2} + f^{2} = 2 c$

$g^{2} + f^{2} + c = 0$

answer is C.

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Detailed Solution

Equation to the pair of tangents drawn from $(0, 0) to x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ is

$(g x + f v + c)^{2} = c (x^{2} + y^{2} + 2 g x + 2 f y + c)$

$\Rightarrow g^{2} x^{2} + 2 f g x y + f^{2} y^{2} + 2 c g x + 2 c f y + c^{2} = c x^{2} + c y^{2} + 2 c g x + 2 f y + c^{2}$

$\Rightarrow (g^{2} - c) x^{2} + 2 f g x y + (f^{2} - c) y^{2} = 0$

The tangents are at right angles $\Rightarrow coeff. of x^{2} + coeff. of y^{2} = 0 \Rightarrow g^{2} - c + f^{2} - c = 0$

$\Rightarrow g^{2} + f^{2} = 2 c$

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