Question

The eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ whose latus rectum is half of its major axis is

• A. $\frac{1}{\sqrt{2}}$
• B. $\sqrt{\frac{2}{3}}$
• C. $\frac{\sqrt{3}}{2}$
• D. None of these

Answer 1

Answer: (A)

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