Question

The eccentricity of an ellipse is $\frac{\sqrt{3}}{2}$ its length of latus reetum is

• A. $\frac{1}{2}$ (length of major axis)
• B. $\frac{1}{3}$ (length of major axis)
• C. $\frac{1}{4}$ (length of major axis)
• D. $\frac{2}{3}$ (length of major axis)

Answer 1

Answer: (C)

$\frac{1}{4}$ (length of major axis)

Eccentricity of ellipse is $e=\frac{c}{a}$

$\therefore \frac{\sqrt{3}}{2}=\frac{c}{a}$ (Given)

$\therefore c=\sqrt{3}$ and $a=2$

By property of ellipse, $c^2=a^2-b^2$

$\therefore {\left(\sqrt{3}\right)}^2={\left(2\right)}^2-b^2$

$\therefore 3=4-b^2$

$\therefore b^2=1$

$\therefore b=1$

Length of latus rectum is given by,

$L=\frac{2b^2}{a}$

$\therefore L=\frac{2{\left(1\right)}^2}{2}$

$\therefore L=1$

Length of major axis is $2a=2\times 2=4$

Thus, we can conclude that,

$L=\frac{1}{4}\left(Lengthofmajoraxis\right)$