Finding Equation of Parabola from 3 Non-Collinear Points

1. Given 3 Non Collinear Points \((x_1,y_1)\), \((x_2,y_2)\) and \((x_3,y_3)\) on a Plane, it is possible to find Equations of 2 Axis Aligned Parabolas, one having Directrix Parallel to \(X\)-Axis, and other having Directrix Parallel to \(Y\)-Axis.
2. The following details the steps for finding out the Explicit Coordinate Equation for Axis Aligned Parabolas passing through 3 Non Collinear Points \((x_1,y_1)\), \((x_2,y_2)\) and \((x_3,y_3)\) using Matrix Inversion.
   
   The Explicit Coordinate Equation for Axis Aligned Parabolas is given by the following
   
   \(y=Ax^2 + Bx + C\)   (For Directrix Parallel to \(X\)-Axis)...(1)
   
   \(x=Ay^2 + By + C\)   (For Directrix Parallel to \(Y\)-Axis)...(2)
   
   Since the Parabola having Directrix Parallel to \(X\)-Axis passes through Points \((x_1,y_1)\), \((x_2,y_2)\) and \((x_3,y_3)\), these Points satisfy the equation of the Parabola. Hence,
   
   \(y_1=A{x_1}^2 + Bx_1 + C\)   ...(3)
   
   \(y_2=A{x_2}^2 + Bx_2 + C\)   ...(4)
   
   \(y_3=A{x_3}^2 + Bx_3 + C\)   ...(5)
   
   The equations (3), (4), and (5) form a set of Linear Equations. These equations can be written in Matrix Multiplication form as
   
   \(\begin{bmatrix}{x_1}^2 & x_1 & 1 \\{x_2}^2 & x_2 & 1 \\ {x_3}^2 & x_3 & 1\end{bmatrix}\begin{bmatrix}A \\ B \\ C\end{bmatrix}=\begin{bmatrix}y_1 \\ y_2 \\ y_3\end{bmatrix}\)
   
   Now, the value of the Co-efficients variables \(A\), \(B\) and \(C\) can be found as follows
   
   \(\begin{bmatrix}A \\ B \\ C\end{bmatrix}={\begin{bmatrix}{x_1}^2 & x_1 & 1 \\{x_2}^2 & x_2 & 1 \\ {x_3}^2 & x_3 & 1\end{bmatrix}}^{-1}\begin{bmatrix}y_1 \\ y_2 \\ y_3\end{bmatrix}\)   ...(6)
   
   Once we get the values of Co-efficients variables \(A\), \(B\) and \(C\) the Explicit Coordinate Equation of the Parabola having Directrix Parallel to \(X\)-Axis can be found out putting these values in equation (1).
   
   Similarly, since the Parabola having Directrix Parallel to \(Y\)-Axis passes through Points \((x_1,y_1)\), \((x_2,y_2)\) and \((x_3,y_3)\), these Points satisfy the equation of the Parabola. Hence,
   
   \(x_1=A{y_1}^2 + By_1 + C\)   ...(7)
   
   \(x_2=A{y_2}^2 + By_2 + C\)   ...(8)
   
   \(x_3=A{y_3}^2 + By_3 + C\)   ...(9)
   
   The equations (7), (8), and (9) form a set of Linear Equations. These equations can be written in Matrix Multiplication form as
   
   \(\begin{bmatrix}{y_1}^2 & y_1 & 1 \\{y_2}^2 & y_2 & 1 \\ {y_3}^2 & y_3 & 1\end{bmatrix}\begin{bmatrix}A \\ B \\ C\end{bmatrix}=\begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix}\)
   
   Now, the values of Co-efficients variables \(A\), \(B\) and \(C\) can be found as follows
   
   \(\begin{bmatrix}A \\ B \\ C\end{bmatrix}={\begin{bmatrix}{y_1}^2 & y_1 & 1 \\{y_2}^2 & y_2 & 1 \\ {y_3}^2 & y_3 & 1\end{bmatrix}}^{-1}\begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix}\)   ...(10)
   
   Once we get the values of Co-efficients variables \(A\), \(B\) and \(C\) the Explicit Coordinate Equation of the Parabola having Directrix Parallel to \(Y\)-Axis can be found out putting these values in equation (2).
3. The following gives Implicit Coordinate Equation of Axis Aligned Parabolas passing through 3 Non Collinear Points \((x_1,y_1)\), \((x_2,y_2)\) and \((x_3,y_3)\) as solution of Determinant Equations
   
   For Parabola having Directrix Parallel to \(X\)-Axis
   
   \(\begin{vmatrix}x^2 & x & y & 1 \\{x_1}^2 & x_1 & y_1 & 1 \\{x_2}^2 & x_2 & y_2 & 1 \\{x_3}^2 & x_3 & y_3 & 1 \end{vmatrix}=0\)   ...(11)
   
   The solution of the Determinant given in above equation (11) gives the following Implicit Coordinate Equation of Axis Aligned Parabola
   
   \(Ax^2 + Bx + Cy + D=0\)
   
   where    \(A=\begin{vmatrix} x_1 & y_1 & 1 \\x_2 & y_2 & 1 \\x_3 & y_3 & 1 \end{vmatrix}\),    \(B=-\begin{vmatrix} {x_1}^2 & y_1 & 1 \\{x_2}^2 & y_2 & 1 \\{x_3}^2 & y_3 & 1 \end{vmatrix}\),    \(C=\begin{vmatrix} {x_1}^2 & x_1 & 1 \\{x_2}^2 & x_2 & 1 \\{x_3}^2 & x_3 & 1 \end{vmatrix}\),    \(D=-\begin{vmatrix} {x_1}^2 & x_1 & y_1 \\{x_2}^2 & x_2 & y_2 \\{x_3}^2 & x_3 & y_3\end{vmatrix}\)
   
   Similarly, for Parabola having Directrix Parallel to \(Y\)-Axis
   
   \(\begin{vmatrix}y^2 & y & x & 1 \\{y_1}^2 & y_1 & x_1 & 1 \\{y_2}^2 & y_2 & x_2 & 1 \\{y_3}^2 & y_3 & x_3 & 1 \end{vmatrix}=0\)   ...(12)
   
   The solution of the Determinant given in above equation (12) gives the following Implicit Coordinate Equation of Axis Aligned Parabola
   
   \(Ay^2 + By + Cx + D=0\)
   
   where    \(A=\begin{vmatrix} y_1 & x_1 & 1 \\y_2 & x_2 & 1 \\y_3 & x_3 & 1 \end{vmatrix}\),    \(B=-\begin{vmatrix} {y_1}^2 & x_1 & 1 \\{y_2}^2 & x_2 & 1 \\{y_3}^2 & x_3 & 1 \end{vmatrix}\),    \(C=\begin{vmatrix} {y_1}^2 & y_1 & 1 \\{y_2}^2 & y_2 & 1 \\{y_3}^2 & y_3 & 1 \end{vmatrix}\),    \(D=-\begin{vmatrix} {y_1}^2 & y_1 & x_1 \\{y_2}^2 & y_2 & x_2 \\{y_3}^2 & y_3 & x_3\end{vmatrix}\)
   
   Please note that Interchaging any number of Rows and Columns in Determinants of equations (11) and (12) DOES NOT Change the value of Determinant and hence DOES NOT Change the Equation of Parabolas.