x²/a²-y²/b²=1,P,A,B是一、双曲线上不同的三点,且AB连线过原点

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  • AB连线过原点,故A点和B点在双曲线的不同分支上,且两坐标关于原点对称,

    设P(x0,y0),A(x1,y1),B(-x1,-y1),

    PB直线的斜率k1=(y0+y1)/(x0+x1),

    PA直线的斜率k2=(y0-y1)/(x0-x1),

    k1*k2=(y0^2-y1^2)/(x0^2-x1^2)=2/3,

    P点坐标代入方程,x0^2/a^2-y0^2/b^2=1,(1),

    A点坐标代入方程,x1^2/a^2-y1^2/b^2=1,(2),

    (1)-(2)式,

    (x0^2-x1^2)/a^2-(y0^2-y1^2)=0,

    b^2/a^2-(y0^2-y2^2)/(x0^2-x2^2)=0,

    b^2/a^2-2/3=0,

    *c^2-a^2)/a^2=2/3,

    (c/a)^2=5/3,

    离心率e=c/a=√15/3.