Question

Write the eccentricity of the hyperbola whose latus-rectum is half of its transverse axis.

Answer 1

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 trasverse 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:

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

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

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

Therefore, the eccentricity is \[\sqrt{\frac{3}{2}}\].