Question

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

• A. $9$
• B. $\frac{13}{12}$
• C. $\frac{32}{3}$
• D. $\frac{64}{3}$

Answer 1

The correct option is B

$\frac{13}{12}$

Step 1. Finding the eccentricity of hyperbola:

Equation of hyperbola

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Given, hyperbola has a length of its conjugate axis equal to $5$ then,

$2b=5$ & its foci is $2ae=13$

therefore,

$$\begin{gathered} 2ae=13 \\ ae=\frac{13}{2}....\left(i\right) \end{gathered}$$& $b=\frac{5}{2}$

Now, we know that,

$\Rightarrow$$\begin{array}{rcl}\frac{25}{4}& =& \frac{169}{4}-a^2\end{array}$ $\begin{array}{rcl}{\mathit{b}}^{\mathbf{2}}& \mathbf{=}& {\mathbf{a}}^{\mathbf{2}}\mathbf{(}{\mathbf{e}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{)}\end{array}$

$\Rightarrow$ $\begin{array}{rcl}{a}^2& =& \frac{144}{4}\end{array}$

$\Rightarrow$ $\begin{array}{rcl}a& =& 6\end{array}$

Step 2. put value of $a$ in equation $\left(i\right)$

$\mathbf{\therefore }\mathit{e}\mathbf{=}\frac{\mathbf{13}}{\mathbf{12}}$

Hence, Correct option is $\left(B\right)$