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

Explanation

Consider the equation of the ellipse is `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
`therefore`Length of major axis `=2a`
Length of minor axis = `2b `
and length of latus rectum=`(2b^(2))/a`
Given that, `(2b^(2))/a=(2b)/2`
`rArr` `a=2b` `rArr` `b=a/2`
We know that, `b^(2)=a^(2)(1-e^(2))`
`rArr (a/2)^(2)=a^(2)(1-e^(2))`
`rArr(a^(2))/4=a^(2)(1-e^(2))`
`rArr1-e^(2)=1/4`
`rArre^(2)=1-1/4`
`therefore e=sqrt(3/4)=sqrt(3/2)`