Let H: x^2 a^2 - y^2 b^2 = 1 be a hyperbola. A circle C is drawn taking the line segment joining the foci of H as its diameter. The directrix of H in the positive x -region intersects the circle C at points P and Q . If the length of the chord PQ is 2 3 times the length of the latus rectum of H , and H passes through the point (3, 2 2 ) , then the value of a^2 + b^2 is:

Step-by-step solution

The foci of H are ( ae, 0) . The circle C with the segment joining the foci as diameter has the equation x^2 + y^2 = a^2e^2 . The directrix of H in the positive x -region is x = a e . To find the intersection of the directrix and the circle, substitute x = a e into the circle's equation: a^2 e^2 + y^2 = a^2e^2 y^2 = a^2e^2 - a^2 e^2 = a^2(e^4 - 1) e^2 y = a e e^4 - 1 The length of the chord PQ is 2|y| = 2a e e^4 - 1 . The length of the latus rectum of H is L = 2b^2 a = 2a^2(e^2 - 1) a = 2a(e^2 - 1) . Given that PQ