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Question Number 80206 by peter frank last updated on 01/Feb/20

Commented by mr W last updated on 01/Feb/20

$t h e l a n g u a g e o f t h e q u e s t i o n i s n o t$ $t o u n d e r s t a n d .$ $p l e a s e m a k e t h e q u e s t i o n c l e a r! w h a t$ $e x a c t i s t o p r o v e ?$
Commented by mr W last updated on 01/Feb/20

	Answered by mr W last updated on 01/Feb/20

	$s a y t h e t a n g e n t p a s s i n g t h e g i v e n p o i n t i s$ $y - \sqrt{a^{2} + b^{2}} = m (x - \frac{a^{2}}{\sqrt{a^{2} - b^{2}}})$ $y = m x + \sqrt{a^{2} + b^{2}} - \frac{{m a}^{2}}{\sqrt{a^{2} - b^{2}}}$ $w e h a v e$ $m^{2} a^{2} + b^{2} = {(\sqrt{a^{2} + b^{2}} - \frac{{m a}^{2}}{\sqrt{a^{2} - b^{2}}})}^{2}$ $m^{2} a^{2} + b^{2} = a^{2} + b^{2} + \frac{m^{2} a^{4}}{a^{2} - b^{2}} - \frac{2 {m a}^{2} \sqrt{a^{2} + b^{2}}}{\sqrt{a^{2} - b^{2}}}$ $\frac{m^{2} b^{2}}{a^{2} - b^{2}} - \frac{2 m \sqrt{a^{2} + b^{2}}}{\sqrt{a^{2} - b^{2}}} + 1 = 0$ $b^{2} m^{2} - 2 m \sqrt{a^{4} - b^{4}} + a^{2} - b^{2} = 0$ $\Rightarrow m = \frac{(\sqrt{a^{2} + b^{2}} \pm a) \sqrt{a^{2} - b^{2}}}{b^{2}}$ $t a n g e n t l i n e 1 :$ $y - \sqrt{a^{2} + b^{2}} = [\frac{(\sqrt{a^{2} + b^{2}} - a) \sqrt{a^{2} - b^{2}}}{b^{2}}] (x - \frac{a^{2}}{\sqrt{a^{2} - b^{2}}})$ $i n t e r s e c t i o n w i t h y - a x i s a t p o i n t P :$ $y_{P} - \sqrt{a^{2} + b^{2}} = [\frac{(\sqrt{a^{2} + b^{2}} - a) \sqrt{a^{2} - b^{2}}}{b^{2}}] (- \frac{a^{2}}{\sqrt{a^{2} - b^{2}}})$ $\Rightarrow y_{P} = (\frac{a^{2}}{b^{2}}) a - (\frac{a^{2}}{b^{2}} - 1) \sqrt{a^{2} + b^{2}}$ $t a n g e n t l i n e 2 :$ $y - \sqrt{a^{2} + b^{2}} = [\frac{(\sqrt{a^{2} + b^{2}} + a) \sqrt{a^{2} - b^{2}}}{b^{2}}] (x - \frac{a^{2}}{\sqrt{a^{2} - b^{2}}})$ $i n t e r s e c t i o n w i t h x - a x i s a t p o i n t Q :$ $0 - \sqrt{a^{2} + b^{2}} = [\frac{(\sqrt{a^{2} + b^{2}} + a) \sqrt{a^{2} - b^{2}}}{b^{2}}] (x_{Q} - \frac{a^{2}}{\sqrt{a^{2} - b^{2}}})$ $\Rightarrow x_{Q} = \frac{a \sqrt{a^{2} + b^{2}} - b^{2}}{\sqrt{a^{2} - b^{2}}}$ $. . . . . .$
	Commented by peter frank last updated on 01/Feb/20

	$t h a n k y o u s i r .$
	Commented by mr W last updated on 01/Feb/20

	$s o l u t i o n i s n o t c o m p l e t e y e t! i t c a n$ $n o t b e c o m p l e t e d s i n c e {i t}^{'} s n o t c l e a r$ $w h a t s h o u l d b e p r o v e d .$ $w h e r e d i d y o u g e t t h e q u e s t i o n ? c a n$ $y o u c h e c k t h e q u e s t i o n o n c e m o r e ?$
	Commented by peter frank last updated on 01/Feb/20

	$I c h e c k t h e q u e s t i o n o n t h e p a p e r i t s e e m s n o p r o b l e m o n t y p i n g m a y$ $l a n g u a g e u s e d$
	Commented by mr W last updated on 01/Feb/20

	Commented by mr W last updated on 01/Feb/20

	$i f y o u u n d e r s t a n d t h i s, p l e a s e e x p l a i n$ $m e w h a t i s m e a n t w i t h i t .$
	Commented by peter frank last updated on 01/Feb/20

	$i t h i n k t h e r e t w o c a s e i n s i d e$ $c a s e 1$ $i n t e r c e p t i o n o f o r d i n a t e$ $t h r o u g h t h e f o c u s$ $c a s e 2$ $i n t e r c e p t o n t h e o r d i n a t e$ $n e a r d i s t a n c e e q u a l t o$ $m a j o r a x i s$
	Commented by mr W last updated on 01/Feb/20

	$i t o t a l l y {d o n}^{'} t u n d e r s t a n d w h a t y o u$ $a r e s a y i n g, m a y b e y o u e i t h e r .$ $w h a t s h o u l d b e p r o v e d ?$ $i h a v e g i v e n a d i a g r a m a b o v e$ $i n t h e c o m m e n t t o t h e q u e s t i o n,$ $t h e d i a g r a m s h o w n t h e s i t u a t i o n .$ $p l e a s e u s e t h i s d i a g r a m f o r y o u r$ $e x p l a n a t i o n . i f y o u a l s o {d o n}^{'} t$ $u n d e r s t a n d t h e q u e s t i o n, j u s t s a y i t,$ $a n d w e f o r g e t t h e q u e s t i o n .$
	Commented by peter frank last updated on 01/Feb/20

	$t o b e h o n e s t s i r i d o n t$ $u n d e r s t a n d . t h a t w h y m y$ $e x p l a n a t i o n w a s o u t o f p o i n t$
	Commented by mr W last updated on 01/Feb/20

	$a l r i g h t, i s e e .$
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