Let an ellipse with its major axis along the x-axis pass through the point (2, 3) . If the sum of the squares of its semi-major and semi-minor axes is 28 , then the eccentricity of the ellipse is

Step-by-step solution

Let the equation of the ellipse be x^2 a^2 + y^2 b^2 = 1 , where a > b . Given that the sum of the squares of the semi-axes is 28 , we have a^2 + b^2 = 28 b^2 = 28 - a^2 . Since the ellipse passes through the point (2, 3) , we substitute x = 2 and y = 3 into the equation: 4 a^2 + 9 28 - a^2 = 1 4(28 - a^2) + 9a^2 = a^2(28 - a^2) 112 - 4a^2 + 9a^2 = 28a^2 - a^4 a^4 - 23a^2 + 112 = 0 Factorising the quadratic in a^2 , we get: (a^2 - 16)(a^2 - 7) = 0 Thus, a^2 = 16 or a^2 = 7 . Since the major axis is along the x-axis