Question

Find the eccentricity of the hyperbola with foci on the $x-axis$ if the length of its conjugate axis is $\frac{3}{4}$ of the length of its transverse axis.

Answer 1

The lengths fo the conjugate axis and the trnverse axis are $2b$ and $2a$, respectively.

We have:

$\begin{array}{c}\Rightarrow 2b=34\times 2a\\ \Rightarrow b=34a\end{array}$

Using the relation $b^2=a^2\left(e^{2-1}\right)$

We get,

$\begin{array}{c}{34}^2{a}^2=a^2\left(e^{2-1}\right)\\ \Rightarrow 916=e^{2-1}\\ e^2=2516\\ e=54\end{array}$

Therefore, the eccentricity of the hyperbola is $54$.