Hyperbola 1 Question 8
9.
The eccentricity of the hyperbola whose length of the latusrectum is equal to $8$ and the length of its conjugate axis is equal to half of the distance between its foci, is
(a) $\frac{4}{3}$
(b) $\frac{4}{\sqrt{3}}$
(c) $\frac{2}{\sqrt{3}}$
(d) $\sqrt{3}$
(2017 Main)
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Answer:
Correct Answer: 9. (c)
Solution:
- We have, $\frac{2 b^{2}}{a}=8$ and $2 b=a e$
$\Rightarrow b^{2}=4 a$ and $2 b=a e$
Consider,
$2 b =a e $
$4 b^{2} =a^{2} e^{2}$
$\Rightarrow 4 a^{2}\left(e^{2}-1\right) =a^{2} e^{2} $
$\Rightarrow 4 e^{2}-4 =e^{2} $ $\quad [\because a \neq 0]$
$\Rightarrow 3 e^{2} =4 $
$\Rightarrow e =\frac{2}{\sqrt{3}}$ $\quad [\because e>0]$
