If the angle between the tangents drawn to the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ $(c > 0)$ from the origin is $\frac{π}{2},$ then

g^{2} + f^{2} = 3 c

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

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

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g^{2} + f^{2} = 4 c

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Solution

The correct option is B $g^{2} + f^{2} = 2 c$
$S_{1} : x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
$∵ c > 0 \Rightarrow (0, 0)$ lies outside to the $S_{1}$
$∵$ tangents from $(0, 0)$ are $⊥$ to each other,
$∴ (0, 0)$ will lie on the director circle of $S_{1}$ (say $S$ )
Hence $S : x^{2} + y^{2} + 2 g x + 2 f y = 0$
As we know that ratios of the radius of $S$ to $S_{1}$ is $\sqrt{2},$
$∴ \frac{\sqrt{g^{2} + f^{2}}}{\sqrt{g^{2} + f^{2} - c}} = \sqrt{2} \Rightarrow 2 c = g^{2} + f^{2}$

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