Question

If the eccentricity of an ellipse is $\frac{5}{8}$ and the distance between its foci is $10$, then find latus rectum of the ellipse.

Answer 1

Given: Eccentricity of an ellipse is $\frac{5}{8}$ and the distance between its foci is $10$

$\Rightarrow 2ae=10$

$[\because$ Distance between foci $=2ae$]

$\Rightarrow ae=5$ ... (i)

$\Rightarrow a=8$ $[\because e=\frac{5}{8}]$

As we know that, $b^2=a^2-a^2{e}^2$

$\Rightarrow b^2=a^2-25$

[ Using equation (i)]

$\Rightarrow a^2-b^2=25$

$\Rightarrow 8^2-b^2=25$

$[\because a=8]$

$\Rightarrow b^2=39$

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

$\Rightarrow$Length of latus rectum$=\frac{2\times 39}{8}=\frac{39}{4}$

$[\because b^2=39,a=8]$

Hence, length of latus rectum is $\frac{39}{4}$