Question

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

Answer 1

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

Distance between foci $=10$

$\Rightarrow 2ae=10$

$\Rightarrow 2a\times \frac{5}{8}=10$

$\Rightarrow a=8$

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

$=\frac{2a^2(1-e^2)}{a}[\because e^2=1-\frac{b^2}{a^2}]$

$=2.8.(1-\frac{25}{64})$

$=\frac{39}{4}$

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