The two circles $x^{2} + y^{2} + 2 g x + c = 0$ and $x^{2} + y^{2} + 2 f y - c = 0$ may touch each other if $g^{2} =$

c

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2 c

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3 c

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4 c

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Solution

The correct option is C $c$
If two circle touches each other then,
$r_{1} + r_{2} = c_{1} c_{2}$
so, $\sqrt{g^{2} - c} + \sqrt{f^{2} + c} = \sqrt{g^{2} + f^{2}}$
$g^{2} + f^{2} + 2 \sqrt{g^{2} - c} \sqrt{f^{2} + c} = g^{2} + f^{2}$
$g^{2} f^{2} + c g^{2} - c f^{2} - c^{2} = 0$
$\Rightarrow g^{2} = \frac{c f^{2} + c^{2}}{f^{2} + c} = c$

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