Converting Parabola Equation from General Parametric to Implicit Coordinate

1. The following are General Parametric Equations of Parabola
   
   \(x=A_1t^2 + B_1t + C_1,\hspace{.5cm}y=A_2t^2 + B_2t + C_2\)
2. The Implicit Coordinate Equation of Parabola is given as the following
   
   \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F=0\)
3. Given the General Parametric Equations of Parabola, the Co-effcients of Implicit Coordinate Equation of Parabola can be found out using the following
   1. Co-efficient of \(x^2\hspace{.3cm}A={A_2}^2\)
   2. Co-efficient of \(xy\hspace{.3cm}B=-2 A_1A_2\)
   3. Co-efficient of \(y^2\hspace{.3cm}C={A_1}^2\)
   4. Co-efficient of \(x\hspace{.3cm}D=B_2\begin{vmatrix}A_2 & B_2 \\ A_1 & B_1\end{vmatrix} - 2 A_2\begin{vmatrix}A_2 & C_2 \\ A_1 & C_1\end{vmatrix}\)
   5. Co-efficient of \(y\hspace{.3cm}E=B_1\begin{vmatrix}A_1 & B_1\\ A_2 & B_2\end{vmatrix} - 2 A_1\begin{vmatrix}A_1 & C_1 \\ A_2 & C_2\end{vmatrix}\)
   6. Constant of Equation \(F =\begin{vmatrix}A_1 & B_1\\ A_2 & B_2\end{vmatrix}\begin{vmatrix}C_1 & B_1\\ C_2 & B_2\end{vmatrix} + {\begin{vmatrix}A_1 & C_1 \\ A_2 & C_2\end{vmatrix}}^2\)
   Hence, the Implicit Coordinate Equation of Parabola is given as
   
   \({A_2}^2x^2 - 2 A_1A_2xy + {A_1}^2y^2 + (B_2\begin{vmatrix}A_2 & B_2 \\ A_1 & B_1\end{vmatrix} - 2 A_2\begin{vmatrix}A_2 & C_2 \\ A_1 & C_1\end{vmatrix})x + (B_1\begin{vmatrix}A_1 & B_1\\ A_2 & B_2\end{vmatrix} - 2 A_1\begin{vmatrix}A_1 & C_1 \\ A_2 & C_2\end{vmatrix})y + (\begin{vmatrix}A_1 & B_1\\ A_2 & B_2\end{vmatrix}\begin{vmatrix}C_1 & B_1\\ C_2 & B_2\end{vmatrix} + {\begin{vmatrix}A_1 & C_1 \\ A_2 & C_2\end{vmatrix}}^2)=0\)