Question

If a hyperbola has length of its conjugate axis equal to $5$ unit and the distance between its foci is $13$ unit, 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 $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
According to question,
$2b=5\Rightarrow b=\frac{5}{2}$
and $2ae=13\Rightarrow ae=\frac{13}{2}$

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