Question

If the length of the latus rectum of an ellipse is equal to half of the length of its minor axis, then its eccentricity is

• A. $\frac{2}{5}$
• B. $\frac{\sqrt{3}}{2}$
• C. $\frac{2}{3}$
• D. $\frac{\sqrt{2}}{3}$

Answer 1

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

Let the equation of ellipse be
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$
Now,
Length of latus rectum $=\frac{2b^2}{a},$
And length of minor axis $=2b$

Given $\frac{2b^2}{a}=b$
$\Rightarrow \frac{b}{a}=\frac{1}{2}(\because b\ne 0)\Rightarrow \frac{b^2}{a^2}=\frac{1}{4}\Rightarrow e=\sqrt{1-\frac{b^2}{a^2}}\therefore e=\frac{\sqrt{3}}{2}$