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  • The eccentricity of the hyperbola whose latus-rectum is 8 and conjugate axis is equal to half the distance between the foci, is   <div class='qna-

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

 

Option: 1

4/3


Option: 2

4 / \sqrt{3}


Option: 3

2 / \sqrt{3}


Option: 4

none of these


Answers (1)

best_answer

\mathrm{\text { We have } \frac{2 b^2}{a}=8 \quad \text { and } 2 b=\frac{1}{2}(2 a e) \quad \therefore \quad \frac{2}{a}\left(\frac{a e}{2}\right)^2=8 \quad \Rightarrow a e^2=16}     ... [i]

\mathrm{\text { Now, } \quad 2 \frac{b^2}{a}=8 \Rightarrow b^2=4 a \Rightarrow a^2\left(e^2-1\right)=4 a \Rightarrow a e^2-a=4}                       ....[ii]

\mathrm{\text { From (i) and (ii), } 16-\frac{16}{e^2}=4 \Rightarrow \frac{16}{e^2}=12 \Rightarrow e=\frac{2}{\sqrt{3}}}

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