Question

Find the equation of the ellipse in the standard form whose distance between foci is $2$ and the length of latus rectum is $\frac{15}{2}$.

Answer 1

Given distance between foci $=2ae=2\Rightarrow ae=1$

And length of latus rectum $=\frac{2b^2}{a}=\frac{15}{2}$

$\Rightarrow \frac{b^2}{a^2}=\frac{15}{4a}=\frac{15e}{4}$

Now use $e^2=1-\frac{b^2}{a^2}$

$\Rightarrow e^2=1-\frac{15e}{4}$

$\Rightarrow 4e^2+15e-4=0$

$\Rightarrow e=\frac{1}{4}$

So $a=4$ and $b=\sqrt{15}$

Hence standard equation of ellipse is given by $\frac{x^2}{16}+\frac{y^2}{15}=1$