Question

The length of the latusrectum of an ellipse is `18/5` and eccentncity is `4/5`, then equation of the ellipse is ______.

Options

• `x^2/25 + "y"^2/8 = 1`

• `x^2/25 + "y"^2/9 = 1`

• `x^2/25 + "y"^2/16 = 1`

• `x^2/16 + "y"^2/9 = 1`

Answer 1

The length of the latusrectum of an ellipse is `18/5` and eccentncity is `4/5`, then equation of the ellipse is `underline(x^2/25 + "y"^2/9 = 1)`.

Explanation:

Given, `(2"b"^2)/"a" = 18/5 => "b"^2/"a" = 9/5`

`=> "b"^2 = 9/5 "a"`    ....(i)

and, e = `sqrt(1 - "b"^2/"a"^2) = 4/5`

`=> 1 - "b"^2/"a"^2 = 16/25`

`=> "b"^2/"a"^2 = 1 - 16/25 = 9/25`

`=> ((9/5)"a")/"a"^2 = 9/25`

`=> 9/"5a" = 9/25`

⇒ a = 5

`therefore "b"^2 = 9/5(5) = 9`  ...[using Eq. (i)]

∴ Required equation of ellipse,

`x^2/25 + "y"^2/9 = 1`