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

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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