Question

Write the eccentricity of an ellipse whose latus-rectum is one half of the minor axis.

Answer 1

Let the equation of ellipse are
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where a>b
According to equation
$\frac{2b^2}{a}=\frac{1}{2}\times \left(2b\right)$
$\Rightarrow 2b^2=ab$
$\Rightarrow \frac{b^2}{b}=\frac{a}{2}$
$\Rightarrow b=\frac{a}{2}$
$\Rightarrow b^2=\frac{a^2}{4}$
Now,
$b^2=a^2(1-e^2)$
$\frac{a^2}{4}=a^2(1-e^2)$
$\Rightarrow \frac{1}{4}=1-e^2$
$\Rightarrow e^2=1-\frac{1}{4}$
$\Rightarrow e^2=\frac{3}{4}$
$\Rightarrow e=\frac{\sqrt{3}}{2}$