Consider the ellipse E given by x^2 18 + y^2 12 = 1 . Let H be the hyperbola whose eccentricity is the reciprocal of the eccentricity of E and whose foci are the same as that of E . Let P and Q be the points of intersection of H and the parabola 5 , y = x^2 in the first quadrant. Let d be the distance between P and Q . If a and b are the integers such that d^2 = a + b 5 , then the value of a - b is __________.

Step-by-step solution

For the ellipse E: x^2 18 + y^2 12 = 1 , we have a_E^2 = 18 and b_E^2 = 12 . The eccentricity e_E is given by: e_E = 1 - b_E^2 a_E^2 = 1 - 12 18 = 1 3 The foci of the ellipse E are ( a_E e_E, 0) = ( 18 1 3 , 0 ) = ( 6 , 0) . For the hyperbola H , the eccentricity e_H is the reciprocal of e_E , so e_H = 3 . Since H has the same foci as E , its foci are ( 6 , 0) . Let the equation of H be x^2 A^2 - y^2 B^2 = 1 . The foci are ( A e_H, 0) , which gives A 3 = 6 A = 2 A^2 = 2 . Also, B^2 = A^2(e_H^2 - 1) = 2(3 - 1) = 4 .