If the eccentricity of an ellipse is `(4)/(9)` and the distance between its foci is 8 units, then find the length of the latus rectum of the ellipse.

Explanation

It is given that `e = (4)/(9)` and distance between the foci of the ellipse i.e., 2c = 8, i.e., c = 4.
We know that c = ae.
`rArr 4 =a((4)/(9))`
`rArr a = (36)/(4)`
`rArr a = 9`
Now, `b^(2) = a^(2)(1-e^(2))`
`rArr b^(2) = 81(1-(16)/(81)) = 81 xx (65)/(81) = 65`
Now the length of the latus rectum is given by `(2b^(2))/(a) i.e., (2(65))/(9) = 14.4` units