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}}$