The eccentricity of the ellipse if the distance between the foci is equal to the length of the latus rectum is a.`(sqrt(5)-1)/2` b. `(sqrt(5)+1)/2` c. `(sqrt(5)-1)/4` d. none of these

Explanation

`frac{2 b^{2}}{a}=2 a e`

`Rightarrow b^{2}=a^{2} e` Now `e=sqrt{1-frac{b^{2}}{a^{2}}}`

`Rightarrow e=sqrt{1-frac{a^{2} e}{a^{2}}}`

`Rightarrow e=sqrt{1-e}`

On squaring both sides

`e^{2}+e-1=0`

`Rightarrow e=frac{-1 pm sqrt{1+4}}{2}`

`Rightarrow e=frac{sqrt{5}-1}{2}`(therefore e cannot be negative )