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Rectangular 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 foci, then $S P \cdot S^{'} P$ is equal to

2

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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 $(C P)^{2}$
Let the coordinate of $P$ be $(x_{1}, y_{1})$
Focal distances $(S P, S^{'} P) = (e x_{1} \pm a)$
Now $(S P) (S^{'} P) = (e x_{1} + a) (e x_{1} - a)$
$= e^{2} x_{1}^{2} - a^{2}$
$= 2 x_{1}^{2} - a^{2}$
$= (x_{1}^{2} - a^{2}) + x_{1}^{2}$
$= y_{1}^{2} + x_{1}^{2} (∵ x_{1}^{2} - y_{1}^{2} = a^{2} \Rightarrow x_{1}^{2} - a^{2} = y_{1}^{2})$
$= (C P)^{2}$

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Rectangular Hyperbola

Standard XII Mathematics

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If P is a point on the rectangular hyperbola x2−y2=a2, C is its centre and S and S′ are foci, then SP⋅S′P is equal to

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