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  • 椭圆(双曲线)与直线联立

    [egin{cases} cfrac{x^2}{a^2} + cfrac{y^2}{b^2} = 1\ y = kx + m end{cases}\ (a^2k^2+b^2)x^2+2kma^2x+a^2m^2-a^2b^2=0\ Delta = 4a^2b^2(A-m^2)=4a^2b^2(a^2k^2+b^2-m^2)\ egin{cases} x_1+x_2 =cfrac{-2kma^2}{a^2k^2+b^2}\ x_1 imes x_2=cfrac{a^2m^2-a^2b^2}{a^2k^2+b^2} end{cases}\ egin{cases} y_1+y_2 =cfrac{2mb^2}{a^2k^2+b^2}\ y_1 imes y_2=cfrac{b^2m^2-a^2b^2k^2}{a^2k^2+b^2} end{cases}\ |AB|=sqrt{(x_1-x_2)^2+(y_1-y_2)^2}=sqrt{1+k^2}|x_1-x_2|=sqrt{1+k^2}sqrt{(x_1+x_2)^2-4x_1x_2}=sqrt{1+k^2}cfrac{sqrt{Delta}}{A}=sqrt{1+k^2}cfrac{2absqrt{A-m^2}}{A} ]

    [egin{cases} cfrac{x^2}{a^2} + cfrac{y^2}{b^2} = 1\ x=ny+m end{cases}\ (a^2n^2+b^2)y^2+2nmb^2x+b^2m^2-a^2b^2=0\ Delta = 4a^2b^2(A-m^2)=4a^2b^2(a^2n^2+b^2-m^2)\ egin{cases} y_1+y_2 =cfrac{-2nmb^2}{a^2n^2+a^2}\ y_1 imes y_2=cfrac{b^2m^2-a^2b^2}{a^2n^2+b^2} end{cases}\ egin{cases} x_1+x_2 =cfrac{2ma^2}{a^2n^2+b^2}\ x_1 imes x_2=cfrac{a^2m^2-a^2b^2n^2}{a^2n^2+b^2} end{cases}\ |AB|=sqrt{(x_1-x_2)^2+(y_1-y_2)^2}=sqrt{1+n^2}|y_1-y_2|=sqrt{1+n^2}sqrt{(y_1+y_2)^2-4y_1y_2}=sqrt{1+n^2}cfrac{sqrt{Delta}}{A}=sqrt{1+n^2}cfrac{2absqrt{A-m^2}}{A} ]

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  • 原文地址:https://www.cnblogs.com/h-hg/p/8485279.html
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