If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is 3 2 units, then its eccentricity is

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Given length of latus-rectum of an ellipse is 2 b 2 a = 4 ⇒ b 2 = 2 a and a - a e = 3 2 ⇒ a 1 - e = 3 2 . Now we know, b 2 = a 2 1 - e 2 ⇒ b 2 = a 1 - e a 1 + e ⇒ 2 a = 3 2 a 1 + e ∵ a 1 - e = 3 2 ⇒ 4 3 = 1 + e ⇒ e = 1 3 .

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