Question

The eccentricity of the hyperbola whose length of latus rectum is half the length of transverse axis is

• A. $\sqrt{\frac{3}{2}}$
• B. $\frac{3}{2}$
• C. $2$
• D. $\frac{8}{3}$

Answer 1

The correct option is A $\sqrt{\frac{3}{2}}$

Let the equation of the hyperbola is
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
For the above hyperbola,
Transverse axis $=2a$
Latus rectum $=\frac{2b^2}{a}$

Given: $\frac{2b^2}{a}=\frac{1}{2}\left(2a\right)$
$\Rightarrow 2b^2=a^2$
$\Rightarrow 2\left(a^2\right(e^2-1\left)\right)=a^2$
$\Rightarrow e^2-1=\frac{1}{2}$
$\Rightarrow e=\sqrt{\frac{3}{2}}$