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Question Number 128892 by n0y0n last updated on 11/Jan/21

Commented by mr W last updated on 11/Jan/21

$r = d i s t a n c e f r o m p e d a l p o i n t (0, 0)$ $t o a p o i n t C (x, y) o n a c u r v e$ $p = d i s t a n c e f r o m p e d a l p o i n t (0, 0)$ $t o t h e t a n g e n t o f t h e c u r v e a t t h e$ $p o i n t C (x, y) .$ $t h e e q u a t i o n w h i c h e x p r e s s e s t h e$ $r e l a t i o n b e t w e e n r a n d p i s c a l l e d$ $t h e p e d a l e q u a t i o n o f t h e c u r v e .$
Commented by bemath last updated on 11/Jan/21

$what is the pedal equation of the ellips$
Commented by mr W last updated on 11/Jan/21

$f o r e l l i p s e i t i s$ $r^{2} + \frac{a^{2} b^{2}}{p^{2}} = a^{2} + b^{2}$
	Answered by BHOOPENDRA last updated on 11/Jan/21

	$t h e e q u a t i o n o f t h e t a n g e n t i s$ $g i v e n$ $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 - - - - (1)$ $i t c a n b e w r i t t e n a s$ $\Rightarrow \frac{x X}{a^{2}} + \frac{y Y}{b^{2}} = 1 - - - - - (2)$ $N o w w e c a n c o m p a r e A X + B y + c = 0$ $w i t h p e r p e n d i c u l a r d i s t a n c e f o r m u l a$ $(0, 0)$ $P = - ∣ \frac{c}{\sqrt{A^{2} + B^{2}}} ∣ {w h e r e A}^{2} = \frac{x}{a^{4}}, B^{2} = \frac{y}{b^{4}}$ $n o w r^{2} = x^{2} + y^{2}$ $\frac{1}{p^{2}} = \frac{x^{2}}{a^{4}} + \frac{y^{2}}{b^{4}} - - - - - (3)$ $n o w a f t e r s i m l i f i c a t i o n$ $\frac{x^{2}}{a^{2}} = \frac{r^{2} - b^{2}}{a^{2} - b^{2}} - - - - - (4)$ $s i m i l a r l y$ $\frac{y^{2}}{b^{2}} = \frac{a^{2} - r^{2}}{a^{2} - b^{2}} - - - - - - (5)$ $n o w p u t v a l u e i n e q u a t i o n (3)$ $w e w i l l g e t$ $\frac{a^{2} b^{2}}{p^{2}} = a^{2} + b^{2} - r^{2}$ $A n d t h i s w i l l b e y o u r p e d a l e q u a t i o n$
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