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. $16 x^{2}+y^{2}+10 x y=2$
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,
$$ \begin{aligned} 2 b & =a e \\ 4 b^{2} & =a^{2} e^{2} \end{aligned} $$
$$ \begin{array}{rlrl} \Rightarrow & & 4 a^{2}\left(e^{2}-1\right) & =a^{2} e^{2} \\ \Rightarrow & 4 e^{2}-4 & =e^{2} \\ \Rightarrow & 3 e^{2} & =4 \\ \Rightarrow & e & =\frac{2}{\sqrt{3}} \end{array} $$
$[\because a \neq 0]$
$[\because e>0]$
