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:
Correct Option: , 2
Solution:
$2 a e=6$ and $\frac{2 a}{e}=12$
$\Rightarrow a e=3$ $\ldots$ (i)
and $\frac{a}{e}=6 \Rightarrow e=\frac{a}{6}$..(ii)
$\Rightarrow a^{2}=18 \quad$ [From (i) and (ii)]
$\Rightarrow \quad b^{2}=a^{2}-a^{2} e^{2}=18-9=9$
$\therefore \quad$ Latus rectum $=\frac{2 b^{2}}{a}=\frac{2 \times 9}{3 \sqrt{2}}=3 \sqrt{2}$
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