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 $\underset{–}{}$

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

$\Rightarrow \frac{2b^2}{a}=10$ [given]

$\Rightarrow b^2=5a$ ⋯(1)

Distance between foci = length of minor axis

$\Rightarrow 2ae=2b$

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

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

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

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

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

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

So, the equation of ellipse is $\frac{x^2}{100}+\frac{y^2}{50}=1$
$\Rightarrow x^2+2y^2=100$