If the eccentricity of an ellipse is 1 2 , then the ratio of the distance between its directrices to the length of its latus rectum is

Step-by-step solution

Let the equation of the ellipse be x^2 a^2 + y^2 b^2 = 1 ( a > b ). The distance between the directrices is given by 2a e . The length of the latus rectum is given by 2b^2 a . The required ratio is 2a e 2b^2 a = a^2 e b^2 . Using the relation b^2 = a^2(1 - e^2) , we get a^2 b^2 = 1 1 - e^2 . Substituting this into the ratio expression, we obtain: Ratio = 1 e(1 - e^2) Given e = 1 2 , the ratio becomes: Ratio = 1 1 2 (1 - 1 4 ) = 1 1 2 3 4 = 1 3 8 = 8 3 Answer: 8 3