The length of the chord of contact with respect to the point on the director circle of circle $x^{2} + y^{2} + 2 a x - 2 b y + a^{2} - b^{2} = 0$ is $k | b |$ units. Then the value of $k$ is

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Solution

$S_{1} : x^{2} + y^{2} + 2 a x - 2 b y + a^{2} - b^{2} = 0, r_{1} = \sqrt{a^{2} + b^{2} - a^{2} + b^{2}} = \sqrt{2} | b |$

From the diagram, $O M P N$ is a square and $P$ lies on director circle of $S_{1} .$
$∴ M N = O P =$ radius of the director circle of $S_{1}$
$\Rightarrow M N = \sqrt{2} | \sqrt{2} b | = 2 | b |$
$∴ k = 2$

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