已知双曲线3X^2-Y^2=3,过P(2,1)点作一直线交双曲线与A、B两点若P为AB中点

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  • 双曲线方程:x^2-y^2/3=1,

    设A(x1,y1),B(x2,y2),P(x0,y0)

    x0=(x1+x2)/2,y0=(y1+y2)/2,

    x0=2,y0=1,

    x1^2-y1^2/3=1,(1)

    x2^2-y2^2/3=1,(2)

    (1)-(2)式,

    1-(1/3)[(y1-y2)/(x1-x2)][(y1+y2)/2]/[(x1+x2)/2]=0.

    其中直线斜率k=(y1-y2)/(x1-x2)

    3-k*1/2=0,

    k=6,

    ∴直线方程为:(y-1)=6(x-2),

    y=6x-11.

    代入双曲线方程,

    3x^2-(6x-11)^2=3,

    33x^2-132x+124=0,

    根据韦达定理,x1+x2=4,

    x1x2=124/33,

    根据弦长公式,

    |AB|=√(1+6^2)(x1-x2)^2=√37*[(x1+x2)^2-4x1x2]

    =√[37*(16-124*4/33)]

    =√(37*32/33)

    =4√2442/33.