椭圆或双曲线的硬解公式

作者：千秋 Lead | 原文：知乎专栏对某椭圆或双曲线与某线的交点

$\displaystyle{(x_1,x_2);(y_1,y_2)}$

$\begin{align} \frac{x^2}{M}+\frac{y^2}{N}&=1 \ Ax+By+C&=0 \end{align}$

有

$\begin{aligned} （MA^2+NB^2+2MAC+M(C^2-N^2B^2)=0\ x_{1}+x_{2}&=-2\frac{MAC}{K} \ y_{1}+y_{2}&=-2\frac{NBC}{K} \x_{1}x_{2}&=\frac{M(C^2-NB^2)}{K} \ y_{1}y_{2}&=\frac{N(C^2-MA^2)}{K} \ x_{1}y_{2}+x_{2}y_{1}&=\frac{2MNAB}{K} \\Delta(x)&=4MNB^2 (K - C^2) \ \Delta(y)&=4MNA^2 (K - C^2) \ \text{其中} \ K&=MA^2+NB^2 \end{aligned}$