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

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10-2. Parabola, Ellipse, Hyperbola

hard

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

$\sqrt 3$

B

$2\sqrt 3$

C

$3\sqrt 2$

D

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

(JEE MAIN-2020)

Solution

Given $2 \mathrm{ae}=6 \Rightarrow \quad \mathrm{ae}=3\dots(1)$

and $\frac{2 a}{e}=12 \Rightarrow \mathbb{a}=6 e\dots(2)$

from $( 1)$ and $( 2)$

$6 e^{2}=3 \Rightarrow \quad e=\frac{1}{\sqrt{2}}$

$\Rightarrow \quad a=3 \sqrt{2}$

Now, $b^{2}=a^{2}\left(1-e^{2}\right)$

$\Rightarrow \mathrm{b}^{2}=18\left(1-\frac{1}{2}\right)=9$

Length of L.R $=\frac{2(9)}{3 \sqrt{2}}=3 \sqrt{2}$

Standard 11

Mathematics

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