Find the equation of the ellipse whose axes are along the coordinate axes, vertices are $(\pm 5, 0)$ and focii at $(\pm 4, 0) .$

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Solution

$\begin{matrix} G i v e n V e r t i c e s (\pm 5, 0) - - - - - (1) sin c e t h e v e r t i c e s a r e o f f o r m (\pm a, 0) - - - (2) h e n c e, M a j o r a x i s i s a l o n g x - a x i s a n d e q u a t i o n o f e l l i p s e i s \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 F r o m (1) & (2) a = 5 a l s o g i v e n c o o r d i n a t e o f f o c i = (\pm 4, 0) w e k n o w t h a t f o c i = (\pm c, 0) s o, c = 4 W e k n o w t h a t c^{2} = a^{2} - b^{2} {(4)}^{2} = {(5)}^{2} - {(b)}^{2} b^{2} = 25 - 16 b^{2} = 9 b = \sqrt{9} ∴ b = 3 H e n c e, b = 3 E q u a t i o n o f e l l i p s e i s \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 P u t t i n g v a l u e \frac{x^{2}}{{(5)}^{2}} + \frac{y^{2}}{{(3)}^{2}} = 1 \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1 W h i c h i s r e q u i r e d e q u a t i o n . \end{matrix}$

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Find the equation of the ellipse whose axes are along the coordinate axes, vertices are(±5,0) and focii at (±4,0).

Find the equation of the ellipse whose axes are along the coordinate axes, vertices are $(\pm 5, 0)$ and focii at $(\pm 4, 0) .$