A variable chord PQ of an ellipse E: x^2 a^2 + y^2 b^2 = 1 ( a > b > 0 ) subtends a right angle at the origin O . The chord PQ always touches a fixed circle of radius 12 5 centered at the origin. If the length of the latus rectum of the ellipse E is 9 2 , then the value of a^2 + 2b^2 is

Step-by-step solution

Let the perpendicular distance from the origin O to the chord PQ be h . Since the chord touches a fixed circle of radius 12 5 centered at the origin, we have h = 12 5 . In the right-angled triangle OPQ , the altitude from the right angle O to the hypotenuse PQ is h . By geometry, we know that: 1 h^2 = 1 OP^2 + 1 OQ^2 Since OP and OQ are mutually perpendicular semi-diameters of the ellipse, they satisfy the property: 1 OP^2 + 1 OQ^2 = 1 a^2 + 1 b^2 Equating the two expressions, we get: 1 a^2 + 1 b^2 = 1 ( 12 5 )^2 =