Given that we need to find the equation of the ellipse whose distance between the foci is 8 units and distance between the directrices is 18 units.

We know that the distance between the foci is 2ae.

⇒ 2ae = 8

⇒ ae = 4 ..... - (1)

We know that the distance between the directrices is .

⇒

⇒ ..... - (2)

(1)×(2)

⇒

⇒ a² = 36 .... (3)

⇒ a = 6

We know that eccentricity of an ellipse is

⇒

⇒ 36 - b² = 16

⇒ b² = 20 ..... (4)

The equation of the ellipse is,

⇒

⇒

⇒ 5x² + 9y² = 180

∴ The equation of the ellipse is 5x² + 9y² = 180.