Question

The eccentricity of the hyperbola whose latus-rectum is half of its transverse axis, is
(a)$\frac{1}{\sqrt{2}}$

(b) $\sqrt{\frac{2}{3}}$

(c) $\sqrt{\frac{3}{2}}$

(d) none of these.

Answer 1

(c) $\sqrt{\frac{3}{2}}$
The lengths of the latus rectum and the transverse axis are $\frac{2b^2}{a}$ and $2a$, respectively. ​
According to the given statement, length of the latus rectum is half of its transverse axis.
$$\begin{gathered} \therefore \frac{2b^2}{a}=\frac{1}{2}\times 2a \\ \Rightarrow \frac{2b^2}{a}=a \\ \Rightarrow 2b^2=a^2 \end{gathered}$$

Eccentricity, $e=\frac{\sqrt{a^2+b^2}}{a}$
Substituting the value $b^2=\frac{a^2}{2}$ , we get:
$$\begin{gathered} e=\frac{\sqrt{a^2+{\displaystyle \raisebox{1ex}{$a^2$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{a} \\ =\frac{a\sqrt{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{a} \\ =\sqrt{\frac{3}{2}} \end{gathered}$$
∴ Eccentricity is $\sqrt{\frac{3}{2}}$