Question

Find the equation of the ellipse with the centre at the origin and whose
(a) foci are $(+4,0)$, and eccentricity is $\frac{2}{3}$,

(b) foci are $(0,\pm 3)$, and eccentricity is $\frac{3}{4}$.

Answer 1

Given foci of ellipse are $(\pm 4,0)$

Eccentricity $=\frac{2}{3},$ center is at origin and

C= distance of foci from origin = 4

We know $e=\frac{c}{a}$

$a=\frac{c}{e}=\frac{4}{2}\times 3=6$

$[a=6]$

and also $c^2=a^2-b^2$

$b^2=36-16$

$b^2=20$

now equation of an ellipse with center (0, 0) us

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

$\frac{x^2}{36}+\frac{y^2}{20}=1$

(b) Given foci of ellipse are $(0,\pm 3)$

the center is at origin and $C=3$

Eccentricity $=\frac{3}{4},$

We know $e=\frac{c}{a}$

$a=\frac{c}{e}=\frac{3}{3}\times 4=4$

$[a=4]$

and also $c^2=a^2-b^2$

$b^2=16-9$

$b^2=7$

now equation of ellipse with center

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

$\frac{x^2}{7}+\frac{y^2}{16}=1$