过四点的抛物线消参

青青子衿

\begin{align*}
\begin{vmatrix}
\left(k x_1+y_1\right){}^2 & x_1 & y_1 & 1 \\
\left(k x_2+y_2\right){}^2 & x_2 & y_2 & 1 \\
\left(k x_3+y_3\right){}^2 & x_3 & y_3 & 1 \\
(k x+y)^2 & x & y & 1 \\
\end{vmatrix}=0\\
\\
\begin{vmatrix}
\left(k x_1+y_1\right){}^2 & x_1 & y_1 & 1 \\
\left(k x_2+y_2\right){}^2 & x_2 & y_2 & 1 \\
\left(k x_3+y_3\right){}^2 & x_3 & y_3 & 1 \\
\left(k x_4+y_4\right){}^2 & x_4 & y_4 & 1 \\
\end{vmatrix}=0
\end{align*}
消去参数k，但是感觉这样的参数k会遗漏某些情形，如何改进呢？

青青子衿

\begin{align*}
P&=\begin{vmatrix}
{x_1}^2 & x_1 & y_1 & 1 \\
{x_2}^2 & x_2 & y_2 & 1 \\
{x_3}^2 & x_3 & y_3 & 1 \\
{x_4}^2 & x_4 & y_4 & 1 \\
\end{vmatrix}\\
\\
Q&=\begin{vmatrix}
x_1y_1 & x_1 & y_1 & 1 \\
x_2y_2 & x_2 & y_2 & 1 \\
x_3y_3 & x_3 & y_3 & 1 \\
x_4y_3 & x_4 & y_4 & 1 \\
\end{vmatrix}\\
\\
\Delta&=\begin{vmatrix}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1 \\
\end{vmatrix}\cdot\begin{vmatrix}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_4 & y_4 & 1 \\
\end{vmatrix}\cdot\begin{vmatrix}
x_1 & y_1 & 1 \\
x_3 & y_3 & 1 \\
x_4 & y_4 & 1 \\
\end{vmatrix}\cdot\begin{vmatrix}
x_2 & y_2 & 1 \\
x_3 & y_3 & 1 \\
x_4 & y_4 & 1 \\
\end{vmatrix}\\
\\
k&=\dfrac{-Q\pm\sqrt{\Delta}}{P}

\end{align*}