Definitions, matrix and main matrix of the conics
Consider the matrix $$\displaystyle \overline{A}= \begin{bmatrix} 2 & 1 & 2 \\ 1 & 1 & 0 \\ 2 & 0 & 3 \end{bmatrix}$$ What is the analytical equation of the conic associated with the previous matrix?
We know that the conic defined by the matrix $\overline{A}$ is given by the equation $(x,y,1)\overline{A}(x,y,1)^T=0$. Therefore, $$\begin{bmatrix} x & y & 1 \end{bmatrix} \begin{bmatrix}2 & 1 & 2 \\ 1 & 1 & 0 \\ 2 & 0 & 3\end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}=\begin{bmatrix} x & y & 1 \end{bmatrix} \begin{bmatrix}2x+y+2 \\ x+y \\ 2x+3\end{bmatrix} =$$ $$=2x^2+xy+2x+xy+y^2+2x+3=2x^2+y^2+2xy+4x+3$$
The analytical equation of the conic is $q(x,y)=2x^2+y^2+2xy+4x+3$