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 length of the conjugate axis and the
transverse axis are $2b$ and $2a$ respectively.
The length of conjugate axis
$=\frac{3}{4}$ [length of transverse axis]

$\Rightarrow 2b=\frac{3}{4}\times 2a$

$\Rightarrow \frac{b}{a}=\frac{3}{4}$

$\Rightarrow \frac{b^2}{a^2}=\frac{9}{16}$

Now, using the relation
$e^2=1+\frac{b^2}{a^2}$

$\Rightarrow e^2=1+\frac{9}{16}$

$\Rightarrow e^2=\frac{25}{16}$

$\Rightarrow e^2=\sqrt{\frac{25}{6}}$

$\Rightarrow e=\frac{5}{4}$

$\because \text{Eccentricity},e=\frac{5}{4}$