Given that we need to find the equation of the ellipse whose eccentricity is , latus - rectum is 5 and centre is at origin.

Let us assume the equation of the ellipse is - - - - (1) (a²>b²) since centre is at origin.

We know that eccentricity of the ellipse is

⇒

⇒

⇒ 9(a² - b²) = 4a²

⇒ 5a² = 9b²

⇒ ..... - - - (2)

We know that length of the latus - rectum is

⇒

⇒

⇒

⇒

⇒

From (2),

⇒

⇒

The equation of the ellipse is

⇒

⇒

⇒

⇒ 20x² + 36y² = 405

∴ The equation of the ellipse is 20x² + 36y² = 405.