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Diameter : Hyperbola

If P is a poi...

Question

If $P$ is a point on the rectangular hyperbola $x^{2} - y^{2} = a^{2}$ , $C$ is its centre and $S,$ and $S^{'}$ are the two foci, then $S P . S^{'} P =$

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$(C P)^{2}$

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$(C S)^{2}$

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$(S S^{'})^{2}$

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Solution

The correct option is B
$(C P)^{2}$

$P S = e x + a = \sqrt{2} x + a$

$P S^{'} = e x - a = \sqrt{2} x - a$ [Here, $e$ is eccentricity]

$P S \times P S^{'} = 2 x^{2} - a^{2}$

$= 2 x^{2} - (x^{2} - y^{2})$

$= x^{2} + y^{2} = (C P)^{2}$

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