Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it ?

Options

A. `(4 sqrt2,2 sqrt2)`
B. `(4 sqrt2,2 sqrt3)`
C. `(4 sqrt3, 2 sqrt3)`
D. `(4sqrt3, 2 sqrt2)`

Answer

D

Explanation

`because (2b ^(2))/(a ) = 8 implies b =4a" "..(i)`
Also `2ae=2b impliesb =ae" "…(ii)`
We know that `b ^(2) =a ^(2) (1-e ^(2)) impliesa ^(2) e ^(2) =a ^(2) (1- e ^(2)) implies 2e ^(2) = 1 implies e = (1)/(sqrt2)`
From (i) `a ^(2) e ^(2) =4a implies a = (4)/(e ^(2)) =8 implies a ^(2) =64`
Also from (i) `b ^(2) =32`
`therefore` Equation ellipse: `(x ^(2))/(64) + (y ^(2))/(32) =1implies x ^(2) + 2y ^(2) =64`
Point `(4 sqrt3, 2 sqrt2)` will satisfy the equation of ellipse