硬解公式

#math

对某椭圆或双曲线与某线

x2M+y2N=1Ax+By+C=0\begin{align} \frac{x^2}{M}+\frac{y^2}{N}&=1 \\ Ax+By+C&=0 \end{align}

有

x1+x2=−2MACKy1+y2=−2NBCKx1x2=M(C2−NB2)Ky1y2=N(C2−MA2)Kx1y2+x2y1=2MNABKΔ(x)=4MNB2(K−C2)Δ(y)=4MNA2(K−C2)其中K=MA2+NB2\begin{align} 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{align}