China Mathematical Competition

Question

In a plane rectangular coordinate system xOy, points A, B are on the parabola y^2 = 4x, satisfying OA OB = -4, and point F is the focus of the parabola. Then S_ OFA S_ OFB = ______.

Accepted Answer

Let F(1, 0), A(x_1, y_1), B(x_2, y_2). Then x_1 = y_1^24, x_2 = y_2^24, and -4 = OA OB = x_1x_2 + y_1y_2 = 116(y_1y_2)^2 + y_1y_2, from which we have 116(y_1y_2 + 8)^2 = 0, or y_1y_2 = -8. Therefore, align* S_ OFA S_ OFB &= ( 12 |OF| |y_1| ) ( 12 |OF| |y_2| ) &= 14 |OF|^2 |y_1 y_2| = 2. align* The answer is 2.