prove that product of lengths of perpendiculars from any point of hyperbola to its asymptotes is constant


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Question Number 43921 by peter frank last updated on 17/Sep/18

$p r o v e t h a t p r o d u c t o f l e n g t h s o f p e r p e n d i c u l a r s$ $f r o m a n y p o i n t o f h y p e r b o l a t o i t s$ $a s y m p t o t e s i s c o n s t a n t$
	Answered by math1967 last updated on 18/Sep/18

	$l e t e q u n . o f h y p e r b o l a i s \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ $e q u n s . o f a s y m p t o t e s a r e b x - a y = 0$ $a n d b x + a y = 0 l e t a n y p t (a s e c \emptyset, b t a n ϕ)$ $p e r p e n d i c u l a r f r o m p t . t o a s y m p t o t e s$ $a r e p_{1}, p_{2} ∴ p_{1} = \frac{b a s e c ϕ - a b t a n ϕ}{\sqrt{b^{2} - a^{2}}}$ $p_{2} = \frac{b a s e c ϕ + a b t a n ϕ}{\sqrt{b^{2} + a^{2}}}$ $∴ p_{1} \times p_{2} = \frac{a^{2} b^{2} ({s e c}^{2} ϕ - {t a n}^{2} ϕ)}{\sqrt{b^{4} - a^{4}}}$ $= \frac{a^{2} b^{2}}{\sqrt{b^{4} - a^{4}}} = c o n s t a n t$
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