A real affine conic section is the zero locus in $\mathbb{R}^2$ of the quadratic form $$q(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f=0.$$
We may understand this as the $Z=1$ affine patch of the locus in the projective plane $P^2(\mathbb{R})$ of the homogeneous quadratic
$$Q(X,Y,Z)=aX^2+2bXY+cY^2+2dXZ+2eYZ+fZ^2\\ =\begin{pmatrix}X & Y & Z\end{pmatrix} M \begin{pmatrix}X \\ Y \\ Z\end{pmatrix},$$
where $$M = \begin{pmatrix}a & b & d\\b & c & e\\d & e & f\end{pmatrix}.$$
One type of discriminant is given by $\Delta_3=\det M,$ the vanishing of which tells us whether the conic section is degenerate.
Another discriminant is found from the degree 2 part of $q,$ found by setting $Z=0:$
$$Q(X,Y,0)=aX^2+2bXY+cY^2\\ =\begin{pmatrix}X & Y\end{pmatrix} N \begin{pmatrix}X \\ Y\end{pmatrix},$$
with $$N = \begin{pmatrix}a & b\\b & c\end{pmatrix}.$$
Then put $\Delta_2=\det N = ac-b^2.$ The sign tells whether the conic is (possibly degenerate) elliptic, parabolic, or hyperbolic. Together these two discriminants classify the conic section, as summarized in the following table:
| -- | $\Delta_3\neq 0$ | $\Delta_3 = 0$ |
|---|---|---|
| $\Delta_2>0$ | ellipse | point |
| $\Delta_2=0$ | parabola | parallel lines or double line |
| $\Delta_2<0$ | hyperbola | intersecting lines |
My question is about the $\Delta_2 = \Delta_3 = 0$ degenerate parabola case. Shouldn't there be a third type of discriminant that distinguishes the two sub-cases, the double line like $x^2=0$ from the parallel lines like $x(x-1)=0$?
Assuming the affine locus doesn't contain the line $y=0$, then this line intersects the degenerate parabola when $ax^2+2dx+f=0,$ so we can check whether it has a double root with the univariate quadratic discriminant $\Delta_1=d^2-af.$ But I guess there's nothing special about the line $y=0$, and if instead we used $x=0,$ we'd get a different discriminant.
What is the right way to find this discriminant? Is there a more invariant or geometric way to see what's going on here?