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)

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

N neha

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