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一、抛物线中的弦中点问题1.已知点A(2,8),B(x_1,y_1),C(x_2,y_2)在抛物线y~2=2px上,△ABC的重心与此抛物线的焦点F重合(如图)(1)写出该抛物线的方程和焦点F的坐标:(2)求线段BC中点M的坐标;(3)求BC所在直线的方程。解:(1)由点A(2,8)在抛物线上y~2=2px,有8~2=2p·2,解得p=16.所以抛物线方程为y~2=32x.焦点F的坐标为(8,0).(2)如图,由于F(8,0)是ΔABC的重心,M是BC的中点,所1以2F
First, the midpoint of the parabola 1. Known points A (2,8), B (x_1, y_1), C (x_2, y_2) on the parabola y ~ 2 = 2px, △ ABC center of gravity and the parabola (1) Write the equation of the parabola and the coordinates of the focal point F: (2) Find the coordinates of the middle point M of the line BC; (3) Find the equation of the line where BC is. Solution: (1) by the point A (2,8) on the parabola y ~ 2 = 2px, 8 ~ 2 = 2p · 2, solution p = 16. So the parabolic equation y ~ 2 = 32x. Focus F The coordinate is (8,0). (2) As shown in the figure, since F (8,0) is the center of gravity of ΔABC and M is the midpoint of BC,