Question

The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through the point $(\sqrt{6},3)$ and the length of the latusrectum is $\frac{18}{5}$. Then, the length of the transverse axis is equal to

• A. $5$
• B. $4$
• C. $3$
• D. $2$
• E. $1$

Answer 1

The correct option is D $2$

Given, equation of hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=\mathrm{1...}\left(i\right)$
Eq. (i) passes through $(\sqrt{6},3)$
$\therefore \frac{{\left(\sqrt{6}\right)}^2}{a^2}-\frac{3^2}{b^2}=1\Rightarrow \frac{6}{a^2}-\frac{9}{b^2}=\mathrm{1...}\left(ii\right)$
$\frac{2b^2}{a}=\frac{18}{5}....\left(iii\right)$
On solving Eqs. (i) and (ii)
$a^2=1,b^2=\frac{9}{5}$