椭圆(双曲线)与直线联立
\[\begin{cases}
\cfrac{x^2}{a^2} + \cfrac{y^2}{b^2} = 1\\
y = kx + m
\end{cases}\\
(a^2k^2+b^2)x^2+2kma^2x+a^2m^2-a^2b^2=0\\
\Delta = 4a^2b^2(A-m^2)=4a^2b^2(a^2k^2+b^2-m^2)\\
\begin{cases}
x_1+x_2 =\cfrac{-2kma^2}{a^2k^2+b^2}\\
x_1\times x_2=\cfrac{a^2m^2-a^2b^2}{a^2k^2+b^2}
\end{cases}\\
\begin{cases}
y_1+y_2 =\cfrac{2mb^2}{a^2k^2+b^2}\\
y_1\times y_2=\cfrac{b^2m^2-a^2b^2k^2}{a^2k^2+b^2}
\end{cases}\\
|AB|=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}=\sqrt{1+k^2}|x_1-x_2|=\sqrt{1+k^2}\sqrt{(x_1+x_2)^2-4x_1x_2}=\sqrt{1+k^2}\cfrac{\sqrt{\Delta}}{A}=\sqrt{1+k^2}\cfrac{2ab\sqrt{A-m^2}}{A}
\]
\[\begin{cases}
\cfrac{x^2}{a^2} + \cfrac{y^2}{b^2} = 1\\
x=ny+m
\end{cases}\\
(a^2n^2+b^2)y^2+2nmb^2x+b^2m^2-a^2b^2=0\\
\Delta = 4a^2b^2(A-m^2)=4a^2b^2(a^2n^2+b^2-m^2)\\
\begin{cases}
y_1+y_2 =\cfrac{-2nmb^2}{a^2n^2+a^2}\\
y_1\times y_2=\cfrac{b^2m^2-a^2b^2}{a^2n^2+b^2}
\end{cases}\\
\begin{cases}
x_1+x_2 =\cfrac{2ma^2}{a^2n^2+b^2}\\
x_1\times x_2=\cfrac{a^2m^2-a^2b^2n^2}{a^2n^2+b^2}
\end{cases}\\
|AB|=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}=\sqrt{1+n^2}|y_1-y_2|=\sqrt{1+n^2}\sqrt{(y_1+y_2)^2-4y_1y_2}=\sqrt{1+n^2}\cfrac{\sqrt{\Delta}}{A}=\sqrt{1+n^2}\cfrac{2ab\sqrt{A-m^2}}{A}
\]
