We have $\;ax^2+by^2+2hxy+2gx+2fy+c=0\;$ . Let us write down this quadratic's expanded matrix:
$$A:=\begin{pmatrix}c&g&f\\g&a&h\\f&h&b\end{pmatrix}$$
Observe that firs row is [free coefficient , half coefficient of x, half coefficient of y], the second row is [hlaf coef. of x, coeff. of x^2 , half coef of xy] and etc.
The corresponding reduced matrix is
$$A':=\begin{pmatrix}a&h\\h&b\end{pmatrix}$$
We have, of course, that both matrices are symmetric. Let us now define for a square matrix $\;B\;$ :
$$\color{red}{r(B)}:=\text{ the rank of}\;\;B\;,\;\;\color{red}{|B|}:=\det B\;,\;\;\color{red}{\sigma(B)}:=\text{ the signature of}\;\;B$$
where signature = the difference between the number of positive elements with the number of negative elements in the main diagonal of the matrix.
Then we have, for example (the whole table is way too long):
$$\begin{cases}r(A)=3\;,\;\;r(A')=2\;,\;\;|A|,A'|>0\;\color{red}\implies\;\text{Empty set}\;(\emptyset)\\{}\\r(A)=3\;,\;\;r(A')=2\;,\;\;|A|<0,\;|A'|>0\;\color{red}\implies\;\text{Ellipse}\\{}\\r(A)=3\;,\;\;r(A')=2\;,\;\;|A'|<0\;\color{red}\implies\;\text{Hyperbola}\\{}\\r(A)=3\,,\,\,r(A')=1\;\color{red}\implies\;\text{Parabola}\end{cases}$$
and etc. The signature kicks in for one pair of parallel lines ($\;\sigma(A)=0\;$ and the ranks one less than the maximal possible), and for empty set ($\;\sigma(A)=2\;$ and the ranks as before)
The above is related to bilinear forms and their close relatives, quadratic forms, Sylvester's Inertia Theorem and etc. It is a little advanced material in linear algebra and it is usually covered, as far as I know, in a second course in this subject at undergraduate level.