Question-46813

Question Number 46813 by peter frank last updated on 31/Oct/18

Commented by peter frank last updated on 02/Nov/18

help please

Answered by ajfour last updated on 04/Nov/18

e q . o f a s y m p t o t e s :

\tan θ = \frac{y}{x} = \pm \frac{b}{a}

l e t e q . o f l i n e P Q R :

y = m (x - h) + k

h = a \sec ϕ, k = b \tan ϕ

m = \tan α

f o r x_{Q} :

\frac{{b x}_{Q}}{a} = m (x_{Q} - h) + k

\Rightarrow x_{Q} = \frac{k - m h}{\frac{b}{a} - m}

r_{1} = (h - x_{Q}) \sec α

= (\frac{\frac{b h}{a} - k}{\frac{b}{a} - m}) \sec α

s i m i l a r l y

r_{2} = (h - x_{R}) \sec α

= (\frac{\frac{b h}{a} + k}{\frac{b}{a} + m}) \sec α

r_{1} r_{2} = [\frac{{(\frac{b h}{a})}^{2} - k^{2}}{{(\frac{b}{a})}^{2} - m^{2}}] \sec^{2} α

= [\frac{\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}}}{\frac{1}{a^{2}} - \frac{m^{2}}{b^{2}}}] (1 + m^{2})

b u t P (h, k) l i e s o n h y p e r b o l a

s o \frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}} = 1, h e n c e

\Rightarrow r_{1} r_{2} = \frac{a^{2} b^{2} (1 + m^{2})}{b^{2} - a^{2} m^{2}} .

Commented by peter frank last updated on 04/Nov/18

thank you sir

Answered by MrW3 last updated on 04/Nov/18

l e t p o i n t P b e (h, k)

\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}} = 1

l i n e t h r o u g h p o i n t P w i t h s l o p e m :

y - k = m (x - h)

e q n . o f a s y m p t o t e 1 :

y = - \frac{b}{a} x

y_{q} = - \frac{b}{a} x_{q}

y_{q} - k = m (x_{q} - h)

\Rightarrow k = - \frac{b}{a} x_{q} - m (x_{q} - h)

\Rightarrow (\frac{b}{a} + m) x_{q} = m h - k

\Rightarrow x_{q} = \frac{a (m h - k)}{m a + b}

\Rightarrow y_{q} = - \frac{b (m h - k)}{m a + b}

e q n . o f a s y m p t o t e 2 :

y = \frac{b}{a} x

y_{r} = \frac{b}{a} x_{r}

y_{r} - k = m (x_{r} - h)

\Rightarrow k = \frac{b}{a} x_{r} - m (x_{r} - h)

\Rightarrow (\frac{b}{a} - m) x_{r} = - (m h - k)

\Rightarrow x_{r} = \frac{a (m h - k)}{a m - b}

\Rightarrow y_{r} = \frac{b (m h - k)}{a m - b}

Q P = \sqrt{{(x_{q} - h)}^{2} + {(y_{q} - k)}^{2}} = \sqrt{{(\frac{a (m h - k)}{m a + b} - h)}^{2} + {(- \frac{b (m h - k)}{m a + b} - k)}^{2}}

\Rightarrow Q P =∣ \frac{(b h + a k) \sqrt{(1 + m^{2})}}{m a + b} ∣

R P = \sqrt{{(x_{r} - h)}^{2} + {(y_{r} - k)}^{2}} = \sqrt{{(\frac{a (m h - k)}{a m - b} - h)}^{2} + {(\frac{b (m h - k)}{a m - b} - k)}^{2}}

\Rightarrow R P =∣ \frac{(b h - a k) \sqrt{(1 + m^{2})}}{m a - b} ∣

Q P \times R P =∣ \frac{(b h + a k) \sqrt{(1 + m^{2})}}{m a + b} \times \frac{(b h - a k) \sqrt{(1 + m^{2})}}{m a - b} ∣

=∣ \frac{(b^{2} h^{2} - a^{2} k^{2}) (1 + m^{2})}{m^{2} a^{2} - b^{2}} ∣

=∣ \frac{(\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}}) a^{2} b^{2} (1 + m^{2})}{m^{2} a^{2} - b^{2}} ∣

= \frac{a^{2} b^{2} (1 + m^{2})}{∣ m^{2} a^{2} - b^{2} ∣}

Commented by peter frank last updated on 04/Nov/18

mrW 3 . thank you