Question

Distance between foci is $8$ and distance between directrices is $6$ of hyperbola, then length of latusrectum is

• A. $4\sqrt{3}$
• B. $\frac{4}{\sqrt{3}}$
• C. $\sqrt{\frac{3}{4}}$
• D. None of these

Answer 1

Answer: (B)

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

Given, $2ae=8$ and $\frac{2a}{e}=6$
$\Rightarrow 2a=6e$
$\therefore e^2=\frac{8}{6}\Rightarrow e^2=\frac{4}{3}$
$\Rightarrow e=\frac{2}{\sqrt{3}}$
$\therefore \frac{4}{\sqrt{3}}a=8\Rightarrow a=2\sqrt{3}$
We know that,
$b^2=a^2(e^2-1)=12(\frac{4}{3}-1)=4$
$\therefore$ Length of latusrectum $=\frac{2b^2}{a}$
$=\frac{2\times 4}{2\sqrt{3}}=\frac{4}{\sqrt{3}}$