If the eccentricity of an ellipse is `5/8` and the distance between its foci is `10 ,` then find the latusrectum of the ellipse.

Explanation

Given that, eccentricity=`5/8i.e.,e=5/8`
Let equation of the ellipse be `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` l since the foci of the ellipse is `)pmae,o)`.
`therefore` Distance between foci =`sqrt((ae+ae)^(2))`
`rArr 2sqrt(a^(2)e^(2))=10` [`because` distance between the foci=10]
`rArr sqrt(a^(2)e^(2))=5`
`rArr a^(2)e^(2)=25`
`rArr a^(2)=(25xx64)/25`
`thereforea=8`
We know that,
`rArr b^(2)=a^(2)(1-e^(2))`
`rArr b^(2)=64(1-25/64)`
`rArr b^(2)=64((64-25)/64)`
`b^(2)=39`
`therefore` Length of latusrectum of ellipse=`(2b^(2))/a=2(39/8)=39/4`