My math teacher told me today that the graph of the equation $ax^2+2hxy+by^2+2gx+2fy+c=0$ can represent more than just a pair of straight lines, as it can be a pair of straight lines only if the determinant:
$\begin{vmatrix}a & h & g\\h & b & f\\g & f & c\end{vmatrix}=0$
Can it also represent graphs other than those of straight lines?
HINT.- You do have $$ax^2+2hxy+by^2+2gx+2fy+c=\begin{bmatrix}x&y&1\end{bmatrix}\begin{bmatrix}a&h&g\\h&b&f\\g&f&c\end{bmatrix}\begin{bmatrix}x\\y\\1\end{bmatrix}$$