Find the equation of the ellipse with ecentricity (1/2) and the focus (2,1) Does the line x=3 touches ellipse.if so at what point?if line x=5 is the line of direction.


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Question Number 51612 by peter frank last updated on 29/Dec/18

$F i n d t h e e q u a t i o n o f$ $t h e e l l i p s e w i t h e c e n t r i c i t y$ $\frac{1}{2} a n d t h e f o c u s (2, 1)$ $D o e s t h e l i n e x = 3 t o u c h e s$ $e l l i p s e . i f s o a t w h a t$ $p o i n t ? i f l i n e x = 5 i s t h e$ $l i n e o f d i r e c t i o n .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 29/Dec/18

	Answered by peter frank last updated on 29/Dec/18

	$\frac{P S}{P M} = e$ $e = \frac{1}{2}$ $P = (x, y)$ $S = (2, 1)$ $M = (5, y) \Rightarrow d i r e c t r i x$ $P S = \sqrt{{(x - 2)}^{2} + {(y - 1)}^{2}}$ $P M = \sqrt{{(x - 5)}^{2} + (y - y)}$ $\frac{P S}{P M} = \frac{1}{2}$ $3 x^{2} + 4 y^{2} - 6 x - 8 y - 5 = 0$ $x = 3$ $y^{2} - 2 y + 1 = 0$ $f r o m$ $b^{2} = 4 a c$ ${(- 2)}^{2} = 4$ $h e n c e x = 3 t o u c h e s t h e$ $e l l i p s e$
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