Question

Find the eccentricity equation of an ellipse whose latus-rectum is
(i) half of its minor axis
(ii) half of its major axis.

Answer 1

Let 2a and 2b the minor and major axes of the ellipse.
(i) When latus-rectum is half of minor axis.
$\frac{2b^2}{a}=\frac{1}{2}\times 2b$
$\Rightarrow 2b^2=ab$
$\Rightarrow \frac{b^2}{b}=\frac{a}{2}$
$\Rightarrow b=\frac{a}{2}$
$\Rightarrow b^2=\frac{a^2}{4}$
Now, $b^2=a^2(1-e^2)$
$\Rightarrow \frac{a^2}{4}=a^2(1-e^2)$
$\Rightarrow \frac{1}{4}=1-e^2$
$\Rightarrow e^2=1-\frac{1}{4}$
$\Rightarrow e^2=\frac{3}{4}$
$\Rightarrow e=\frac{\sqrt{3}}{2}$
(ii) When latus-rectum is half of major-axis.
$\frac{2b^2}{a}=\frac{1}{2}\times 2a$
$\Rightarrow 2b^2=a^2$
$\Rightarrow a^2=2b^2$
Now,
$\Rightarrow b^2=a^2(1-e^2)$
$\Rightarrow b^2=2b^2(1-e^2)$
$\Rightarrow 1=2(1-e^2)$
$\Rightarrow 1=2-2e^2$
$\Rightarrow 2e^2=2-1$
$\Rightarrow e^2=\frac{1}{2}$
$\Rightarrow e=\frac{1}{\sqrt{2}}$