Question

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

Answer 1

$\text{Let 2a and 2b be the transverse and conjugate axes and e be the eccentricit.}$

$Then,$

$\text{The length of conjugate aaxis}=\frac{3}{4}\left[lengthoftransverseaxis\right]$

$\Rightarrow 2b=\frac{3}{4}\times \left(2a\right)$

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

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

$\text{Now,using the relation}$

$e=\sqrt{1+\frac{b^2}{a^2}}$

$=\sqrt{1+\frac{9}{16}}$

$=\sqrt{\frac{25}{16}}$

$=\frac{5}{4}$

$Hence,e=\frac{5}{4}$