Let P be a point in the first quadrant lying on a hyperbola H centered at the origin. If the difference of the focal distances of the point P is 4 , the distance between the foci of H is 8 , and the product of the focal distances of P is 20 , then the square of the distance of P from the origin is equal to

Step-by-step solution

Let the equation of the hyperbola be x^2 a^2 - y^2 b^2 = 1 . The difference of the focal distances of any point on the hyperbola is 2a . 2a = 4 a = 2 The distance between the foci is 2ae . 2ae = 8 ae = 4 e = 2 Using the relation b^2 = a^2(e^2 - 1) : b^2 = 4(4 - 1) = 12 The product of the focal distances of a point P (x, y) on the hyperbola is given by e^2 x^2 - a^2 . e^2 x^2 - a^2 = 20 4x^2 - 4 = 20 4x^2 = 24 x^2 = 6 Since P lies on the hyperbola, substitute x^2 = 6 into the hyperbola equation: 6 4 - y^2 12 = 1 3 2