已知抛物线y^2=4x 准线与x轴交于N 过N做斜率为k的直线交抛物线于A,B 且 向量AB=2倍向量NA,求k=?

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  • y^2=4x=2px,p=2,故准线方程是x=-p/2=-1,即N(-1,0)

    A(X1,Y1),B(X2,Y2)

    向量AB=(x2-x1,y2-y1),NA=(x1+1,y1)

    AB=2NA

    即有y2-y1=2y1,x2-x1=2x1+2

    y2=3y1

    x2=3x1+2

    y2^2=4x2

    y1^2=4x1

    y2^2/y1^2=x2/x1=9

    x2=9x1

    9x1=3x1+2

    x1=1/3, y1=2/3根号3或-2/3根号3

    x2=3, y2=2根号3或-2根号3

    K=(y2-y1)/(x2-x1)=(2根号3-2/3根号3)/(3-1/3)=根号3/2

    或者说=(-2根号3+2/3根号3)/(3-1/3)=-根号3/2