Euclidean invariants of the conics

The associated matrix is $$\displaystyle \overline{A}=\begin{bmatrix} 2 & 2 & 1 \\ 2 & 1 & 0 \\ 1 & 0 & 4 \end{bmatrix}$$ The associated euclidean invariants are: $$D_3=det \overline{A}=8-1-16=-9$$ $$d_2=2-4=-2$$ $$d_1=2+1=3$$ We do not compute $D_2$ because the determinant of the associated matrix is other than zero.

From the classification scheme, as $D_3\neq0$ and $d_2 < 0$, the conic is a hyperbola.