A circle C centered at the origin passes through the foci of a hyperbola H : x^2 a^2 - y^2 b^2 = 1 ( a > 0, b > 0 ). The circle C intersects the hyperbola H at a point P in the first quadrant. If the product of the focal distances of the point P is 18 and the sum of the focal distances of P is 12 , then the square of the length of the latus rectum of H is equal to

Step-by-step solution

Let the foci of the hyperbola be S(ae, 0) and S'(-ae, 0) . Since the circle C is centered at the origin and passes through the foci, its radius is ae . For the point P (x, y) on the circle, the square of its distance from the origin is: x^2 + y^2 = a^2 e^2 Since P also lies on the hyperbola, we have x^2 a^2 - y^2 b^2 = 1 y^2 = b^2 ( x^2 a^2 - 1 ) . Substituting this into the circle's equation: x^2 + b^2 x^2 a^2 - b^2 = a^2 e^2 x^2 ( 1 + b^2 a^2 ) - b^2 = a^2 e^2 Since e^2 = 1 + b^2 a^2 , this becomes: x^2 e^2 - b^2