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If the straight line $x \cos \alpha+y \sin \alpha=p$ be a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then
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$a^2 \cos ^2 \alpha-b^2 \sin ^2 \alpha=p^2$
$x \cos \alpha+y \sin \alpha=p \Rightarrow y=-\cot \alpha . x+p \operatorname{cosec} \alpha$
It is tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ Therefore, $p^2 \operatorname{cosec}^2 \alpha=a^2 \cot ^2 \alpha-b^2$ $\Rightarrow a^2 \cos ^2 \alpha-b^2 \sin ^2 \alpha=p^2$.
It is tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ Therefore, $p^2 \operatorname{cosec}^2 \alpha=a^2 \cot ^2 \alpha-b^2$ $\Rightarrow a^2 \cos ^2 \alpha-b^2 \sin ^2 \alpha=p^2$.
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