已知A(-1,0)B(1,-1)抛物线C:y2=4x,O为原点,过点A的动直线l交抛物线C于M,P,直线MB交抛物线C于

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  • 设点M(y²1/4,y1).P(y²2/4,y2) A、M、P三点共线

    Kam=Kmp

    y1/(y²1/4+1)=(y1-y2)/(y²1/4-y²2/4)化简整理得y1y2=4 (1)

    设点Q(y²3/4,y3) M.B.Q三点共线

    Kbq=Kmq

    (y3+1)/(y²3/4-1)=(y3-y1)/(y²3/4-y²1/4)化简整理得y1y3+y1+y3+4=0 (2)

    由(1)得y1=4/y2代入(2)得4y3/y2+4/y2+y3+4=0

    整理得 4(y2+y3)+y2y3+4=0 (3)

    Kpq=(y2-y3)/(y²2/4-y²3/4)=4/(y2+y3)

    直线PQ的方程是y-y2=4/(y2+y3)(x-y²2/4)

    (y-y2)(y2+y3)=4x-y²2

    y(y2+y3)-y2y3=4x

    将(3)代入整理得

    y(y2+y3)+4(y2+y3)=4x-4

    (y+4)(y2+y3)=4(x-1)

    (y+4)=[4/(y2+y3)](x-1)

    直线PQ恒过一个定点(1,-4)