Question

The eccentricity of the hyperbola whose latus-rectum is half of its transverse axis, is 

Options

• \[\frac{1}{\sqrt{2}}\]

• \[\sqrt{\frac{2}{3}}\]

• \[\sqrt{\frac{3}{2}}\]

•  none of these.

Answer 1

\[\sqrt{\frac{3}{2}}\]

The lengths of the latus rectum and the transverse axis are \[\frac{2 b^2}{a}\] and \[2a\] , respectively. ​
According to the given statement, length of the latus rectum is half of its transverse axis.

\[\therefore \frac{2 b^2}{a} = \frac{1}{2} \times 2a\]

\[ \Rightarrow \frac{2 b^2}{a} = a\]

\[ \Rightarrow 2 b^2 = a^2 \]

Eccentricity, 

\[e = \frac{\sqrt{a^2 + b^2}}{a}\]

Substituting the value

\[b^2 = \frac{a^2}{2}\], we get:

\[e = \frac{\sqrt{a^2 + \frac{a^2}{2}}}{a}\]

\[ = \frac{a\sqrt{\frac{3}{2}}}{a}\]

\[ = \sqrt{\frac{3}{2}}\]

∴ Eccentricity is \[\sqrt{\frac{3}{2}}\]