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The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is :

Option 1)
$\frac{4}{3}$

Option 2)
$\frac{4}{\sqrt{3}}$

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

Option 4)
$\sqrt{3}$

Answers (2)

best_answer

As we learnt in

Length of latus Rectum -

$\frac{2b^{2}}{a}$

- wherein

For the Hyperbola

$\frac{x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}= 1$

given kength of LR=8

Hence $\frac{2b^{2}}{a}=8 - - -- - - - -\left ( 1 \right )$

and $2b=\frac{1}{2}\left ( 2ae \right ) - -- - -- - - - - \left ( 2 \right )$

from (1) and (2)

$\frac{2}{a}\times \frac{a^{2}e^{2}}{4}=8$

$ae^{2}=16- - - -- - - - \left ( 3 \right )$

Also, $b^{2}=4a$

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

hence $a^{2}e^{2}-a^{2}=4a$

$\Rightarrow ae^{2}-a=4$

from (3)

16-a=4 $\Rightarrow$ a=12

$ae^{2}=16 \Rightarrow e^{2}=\frac{16}{12}=\frac{4}{3}$

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

Option 1)

$\frac{4}{3}$

Incorrect option

Option 2)

$\frac{4}{\sqrt{3}}$

Incorrect option

Option 3)

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

Correct option

Option 4)

$\sqrt{3}$

Incorrect option

Posted by

divya.saini

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