Find the equation of an ellipse the distance between the foci is 8 units and the distance between the directrices is 18 units.

Options

A. `5x^(2) - 9y^(2) = 180`
B. `9x^(2) + 5y^(2) = 180`
C. `x^(2) + 9y^(2) = 180`
D. `5x^(2) + 9y^(2) = 180`

Answer

D

Explanation

Let the equation of the ellipse be `x^(2)/a^(2) + y^(2)/b^(2) = 1`.
We have,
`2ae = 8 and (2a)/(e) = 18`
`rArr ae = 4 and a = 9e rArr e = 2/3 and a = 6`
`therefore b^(2) = a^(2) (1 - e^(2)) rArr b^(2) = 36 (1 - 4/9) = 20`
Hence, the equation of the ellipse is
`x^(2)/36 + y^(2)/20 = 1 rArr 5x^(2) + 9y^(2) = 180`.