Question

The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ has its conjugate axis of length $5$ and passes through the point $(2,1)$. The length of latus rectum is :

• A. $\frac{5}{4}\sqrt{29}$
• B. $\frac{5}{8}\sqrt{29}$
• C. $\sqrt{29}$
• D. $\frac{\sqrt{29}}{4}$

Answer 1

The correct option is A $\frac{5}{4}\sqrt{29}$

$b=\frac{5}{2}$
$(2,1)$ lies on $\frac{x^2}{a^2}-\frac{4y^2}{25}=1$
$\therefore \frac{4}{a^2}=\frac{29}{25}$

$\Rightarrow a=\frac{10}{\sqrt{29}}$
Latus rectum $\frac{2b^2}{a}=2\left(\frac{25}{4}\right).\frac{\sqrt{29}}{10}$

$=\frac{5}{4}\sqrt{29}$