Question

If the latus rectum of the ellipse with axis along $x-$axis and centre at origin is $10,$ distance between foci = length of minor axis, then the equation of the ellipse is ______.

Answer 1

As axis is along $x-$ axis and centre is at origin, so the equation of ellipse is of form

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

Length of latus rectum $=\frac{2b^2}{a}$

$\Rightarrow \frac{2b^2}{a}=10\left[givne\right]$

$\Rightarrow b^2=5a\dots \left(1\right)$

Distance between foci = length of minor axis

$\Rightarrow 2ae=2b$

$\Rightarrow a^2{e}^2=b^2$

$\Rightarrow a^2(1-\frac{b^2}{a^2})=b^2$

$\Rightarrow a^2=2b^2$

$\Rightarrow a^2=10a\text{[From equation (1)]}$

$\therefore a=10[\because a\ne 0]$

$\Rightarrow b^2=5a=50$

So, the equation of ellipse is

$\frac{x^2}{100}+\frac{y^2}{50}=1$

$\Rightarrow x^2+2y^2=100$