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

Explanation

equation of ellipse be `frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1(a>b)`

Length of latus rectum `=frac{2 b^{2}}{a}`,

and length of minor axis `=2 b`.

Given, `frac{2 b^{2}}{a}=b`

`Rightarrow frac{b}{a}=frac{1}{2} quad(therefore b ne 0) `

`Rightarrow frac{b^{2}}{a^{2}}=frac{1}{4} `

`therefore e=sqrt{1-frac{b^{2}}{a^{2}}}=frac{sqrt{3}}{2}`