Question

Find the equation of the ellipse given that its eccentricity is 2/3 and latus rectum is 5 and centre is (0,0).

Answer 1

Eccentricity, e = $\frac{2}{3}$

Length of latus rectum = 5

Centre (0,0)

$e=\frac{\sqrt{a^2-b^2}}{a}=\frac{2}{3}$

$\frac{2b^2}{a}=5$

$b^2=\frac{5a}{2}$

Therefore,

$\frac{2a^2-5a}{2a^2}=\frac{4}{9}$

$10a^2-45a=0$

$a=0$ or $\frac{9}{2}$

Since, $a\ne 0$

So, $a=\frac{9}{2}$

$a^2=\frac{81}{4}$

$b^2=\frac{45}{4}$

Therefore, equation of ellipse is,

$\frac{4x^2}{81}+\frac{4y^2}{45}=1$