Find the eccentricity of an ellipse, if the length of its latus rectum is 4 units and distance between its vertex and the nearest focus is (3 / 2 ) units.

Step-by-step solution

Let the equation of an ellipse ( x^2 a^2 + y^2 b^2 =1,(a > b) ) Then according to the question, Length of latus rectum (= 2 b^2 a =4 ) ...(i) and ( a-a e= 3 2 a(1-e)= 3 2 ) ...(ii) From Eqs. (i) and (ii), we get ( aligned 2 b^2 a a(1-e) & = 4 3 2 b^2 a^2 = 8 3 (1-e) (1-e^2 ) & = 4 3 (1-e) b^2 a^2 =1-e^2 1+e & = 4 3 e= 4 3 -1 e= 1 3 aligned ) Hence, option (a) is correct.