Question

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 :

• A. \(\frac{2}{\sqrt3}\)
• B. \(\sqrt3\)
• C. \(\frac43\)
• D. \(\frac{4}{\sqrt3}\)

Hint:

Use \(2b=ae\) (conjugate axis is half of \(2ae\)) together with \(b^2=a^2(e^2-1)\).

Answer 1

The Correct Option is A

Let the hyperbola be \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\). Latus rectum \(=\frac{2b^2}{a}=8\). Conjugate axis \(=2b\); distance between foci \(=2ae\).
Given \(2b=\frac12(2ae)=ae\), so \(b=\frac{ae}{2}\) and \(b^2=\frac{a^2e^2}{4}\).
Also \(b^2=a^2(e^2-1)\). Hence \(\frac{e^2}{4}=e^2-1\Rightarrow e^2=4e^2-4\Rightarrow e^2=\frac43\).
\(e=\frac{2}{\sqrt3}\). (The latus rectum condition only fixes \(a\): \(\frac{2b^2}{a}=\frac{ae^2}{2}=8\Rightarrow a=12\).)
Answer: (1) \(\frac{2}{\sqrt3}\)