For an ellipse with its major axis along the x-axis and center at the origin, the distance between the foci, the length of the latus rectum, and the length of the major axis are in an arithmetic progression in that order. If the ellipse passes through the point (1, 3 ) , then the square of the length of its minor axis is equal to

Step-by-step solution

Let the equation of the ellipse be x^2 a^2 + y^2 b^2 = 1 with a > b . The distance between the foci is 2ae . The length of the latus rectum is 2b^2 a = 2a^2(1-e^2) a = 2a(1-e^2) . The length of the major axis is 2a . Since these three quantities are in an arithmetic progression, we have: 2 ( 2a(1-e^2) ) = 2ae + 2a Dividing by 2a (since a 0 ): 2(1-e^2) = e + 1 2(1-e)(1+e) = e + 1 Since e -1 , dividing by e+1 gives: 2(1-e) = 1 1-e = 1 2 e = 1 2 Now, b^2 = a^2(1-e^2) = a^2 (1 - 1 4 ) = 3a^2 4 . The equation of the ell