Question

If the distance between the foci of an ellipse is $6$ and the distance between its directrices is $12$, then the length of its latus rectum is

• A. $2\sqrt{3}$
• B. $\sqrt{3}$
• C. $\frac{3}{\sqrt{2}}$
• D. $3\sqrt{2}$

Answer 1

The correct option is D $3\sqrt{2}$

Let the equation of ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$
Now, $2ae=6$ and $\frac{2a}{e}=12$
$\Rightarrow ae=3$ and $\frac{a}{e}=6$
$\Rightarrow a^2=18$

$a^2{e}^2=c^2=a^2-b^2=9\Rightarrow b^2=9$
Length of latus rectum $=\frac{2b^2}{a}=\frac{2\times 9}{\sqrt{18}}=3\sqrt{2}$