Given that we need to find the eccentricity of an ellipse.

(i) If latus - rectum is half of its minor axis

We know that the length of the semi - minor axis is b and the length of the latus - rectum is .

⇒

⇒ a = 2b .... (1)

We know that eccentricity of an ellipse is

From (1)

⇒

⇒

⇒

⇒

⇒ .

(ii) If latus - rectum is half of its major axis

We know that the length of the semi - major axis is a and the length of the latus - rectum is .

⇒

⇒ a² = 2b² .... (1)

We know that eccentricity of an ellipse is

From (1)

⇒

⇒

⇒

⇒

⇒ .