Question

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

Answer 1

$\text{Let the equation of hyperbola be}$

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$Accordingtoquestion$

$Lengthoflatus-rectum=\frac{1}{2}$

$\left[lengthoftransverseaxis\right]$

$\Rightarrow \frac{2b^2}{a}=\frac{1}{2}\times 2a$

$\Rightarrow \frac{b^2}{a^2}=\frac{1}{2}$

$Now,$

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

$e=\sqrt{1+\frac{1}{2}}$

$=\sqrt{\frac{3}{2}}$

$\therefore e=\sqrt{\frac{3}{2}}$