The equations of two ellipses are be x^2 4 + y^2 2 =1 and x^2 36 + y^2 b^2 =1 . If the product of their eccentricities is 2 3 , then the product of the length of the major axis and minor axis of the second ellipse is

Step-by-step solution

Given: Two ellipses E₁: x^2 4 + y^2 2 = 1 and E₂: x^2 36 + y^2 b^2 = 1 with eccentricity product e₁ e₂ = 2 3 . The eccentricity of E₁ is e₁ = 1 - b₁^2 a₁^2 = 1 - 2 4 = 1 2 . Since e₁ e₂ = 2 3 , then e₂ = 2 3 . Case 1: Major axis along x-axis ( b^2 e₂ = 1 - b^2 36 = 2 3 1 - b^2 36 = 4 9 b^2 36 = 5 9 b^2 = 20 Major axis length 2a₂ = 12 , minor axis length 2b₂ = 4 5 Product: 12 4 5 = 48 5 Case 2: Major axis along y-axis ( b^2 > 36 ) e₂ = 1 - 36 b^2 = 2 3 1 - 36 b^2 = 4 9 36 b^2 = 5 9 b^2 = 324 5 Axis product 432 5 5 n