Question

Let $P$ be any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that the absolute difference of the distances of $P$ from the two foci is $12.$ If the eccentricity of the hyperbola is $2,$ then the length of the latus rectum is

• A. $4\sqrt{3}$ unit
• B. $18$ unit
• C. $2\sqrt{3}$ unit
• D. $36$ unit

Answer 1

The correct option is D $36$ unit

Absolute difference of the distances of $P$ from the two foci is equal to the length of transverse axis.
$\Rightarrow 2a=12$
$\Rightarrow a=6$

Given, $e=2$
$\Rightarrow \sqrt{1+\frac{b^2}{a^2}}=2$
$\Rightarrow b^2=108$
Length of latus rectum $=\frac{2b^2}{a}=36$