If the eccentricity of the ellipse x^2 a^2 + y^2 b^2 = 1, (a > b) is 2 3 and its focal chord is 3x + 2y - 6 = 0 , then the value of a^2 + b^2 is...

Step-by-step solution

The equation of the given ellipse is x^2 a^2 + y^2 b^2 = 1 with a > b . The foci of the ellipse are ( ae, 0) . Since the line 3x + 2y - 6 = 0 is a focal chord, it must pass through one of the foci. Substituting y = 0 in the equation of the line: 3x + 0 - 6 = 0 x = 2 Thus, the focus is (2, 0) , which gives ae = 2 . Given the eccentricity e = 2 3 , substituting this value yields: a ( 2 3 ) = 2 a = 3 Squaring both sides, a^2 = 9 . For an ellipse, the relation between a, b, and e is given by b^2 = a^2(1 - e^2) . Substi