Length of the tangent drawn from any point on the circle $x^{2} + y^{2} + 2 g x + 2 f y + c_{1} = 0$ to the circle is $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$

\sqrt{c_{1} - c}

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\sqrt{c - c_{1}}

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\sqrt{c_{1} + c}

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None of these

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Solution

The correct option is B $\sqrt{c - c_{1}}$
Suppose $(x_{1}, y_{1})$ be any point on first circle from which tangent is to be drawn, then
$x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c_{1} = 0$ ….(i)
and also length of tangent
$= \sqrt{S_{2}} = \sqrt{x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c}$ ….(ii)
From (i), we get (ii) as $\sqrt{c - c_{1}} .$

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