Question

Find the eccentricity of the hyperbola, the length of whose conjugate axis is $\frac{3}{4}$ of the length of transverse axis.

Answer 1

The lengths of the conjugate axis and the transverse axis are $2b$ and $2a$, respectively.
We have:
$$\begin{gathered} \Rightarrow 2b=\frac{3}{4}\times 2a \\ \Rightarrow b=\frac{3}{4}a \end{gathered}$$
Using the relation $b^2=a^2(e^2-1)$, we get:

$$\begin{gathered} \Rightarrow {\left(\frac{3}{4}\right)}^2{a}^2=a^2(e^2-1) \\ \Rightarrow \frac{9}{16}=e^2-1 \\ \Rightarrow e^2=\frac{25}{16} \\ \Rightarrow e=\frac{5}{4} \end{gathered}$$
Therefore, the eccentricity of the hyperbola is $\frac{5}{4}$.