First slide

Ellipse

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:

Easy

A

$\sqrt{3}$
B

$3 \sqrt{2}$
C

$\frac{3}{\sqrt{2}}$
D

$2 \sqrt{3}$

Solution

$2 a e = 6 a n d \frac{2 a}{e} = 12$
$\Rightarrow a e = 3 a n d \frac{a}{e} = 6$
$\Rightarrow a^{2} = 18$
$\Rightarrow b^{2} = a^{2} - a^{2} e^{2} = 18 - 9 = 9 ∴ L e n g t h o f l a t u s r e c t u m = \frac{2 b^{2}}{a} = \frac{2 \times 9}{3 \sqrt{2}} = 3 \sqrt{2}$

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