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

Engineering

Mathematics

Standard Hyperbola

Question

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

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

$\frac{4}{3}$

$\sqrt{3}$

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2a = 8 ⇒ a = 4

2b = $\frac{1}{2}$ (2ae) ⇒ 2b = 4e ⇒ b = 2e

e² = 1 + $\frac{b^{2}}{a^{2}} = 1 + \frac{4 e^{2}}{16}$

$\Rightarrow \frac{3 e^{2}}{4} = 1 \Rightarrow e = \frac{2}{\sqrt{3}}$