Question

If a hyperbola has length of its conjugate axis equal to $5$ and the distance between its foci is $13$, then the eccentricity of the hyperbola is :

• A. $\frac{13}{6}$
• B. $2$
• C. $\frac{13}{8}$
• D. $\frac{13}{12}$

Answer 1

The correct option is D $\frac{13}{12}$

Let equation of the hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
$2b=5\Rightarrow b=\frac{5}{2}$
and $2c=13\Rightarrow c=\frac{13}{2}$

$c^2=a^2+b^2$
$\Rightarrow {\left(\frac{13}{2}\right)}^2=a^2+{\left(\frac{5}{2}\right)}^2$
$\Rightarrow a=6$
Hence, eccentricity, $e=\frac{c}{a}=\frac{13}{12}$