A parabola P is given by the equation y^2 = 8 2 x . A hyperbola H is defined such that its foci lie at the focus and the foot of the directrix of P . If H passes through the point (3, 1) , then the value of 3(L^2 + e^2) is (where L is the length of the latus rectum of H and e is its eccentricity):

Step-by-step solution

For the parabola P: y^2 = 8 2 x , comparing with y^2 = 4Ax , we get A = 2 2 . The focus of P is (2 2 , 0) and the foot of its directrix is (-2 2 , 0) . These are the foci of the hyperbola H . Let the equation of H be x^2 a^2 - y^2 b^2 = 1 . The foci of H are ( ae, 0) , so ae = 2 2 . Using the relation b^2 = a^2(e^2 - 1) , we get a^2e^2 = a^2 + b^2 . Thus, a^2 + b^2 = (2 2 )^2 = 8 , which gives b^2 = 8 - a^2 . The equation of H becomes x^2 a^2 - y^2 8 - a^2 = 1 . Since H passes through (3, 1) , we substitute x = 3,