= Conic type discriminant
{title2=$\Delta_2=B^2-4AC$}
For a second-degree equation $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$, the number $\Delta_2=B^2-4AC$ is the <discriminant of a binary quadratic form> associated with its quadratic part. For a nonempty, nondegenerate real <conic section>, a negative value gives an <ellipse>, zero a <parabola>, and a positive value a <hyperbola>. It is invariant under a <rotation> of orthonormal coordinates.<Precalculus 2e by Jay Abramson/10.4 Rotation of Axes>[Precalculus 2e, §10.4, Identifying Conics without Rotating Axes and Using the Discriminant to Identify a Conic]{ref}
A short explanation uses the <symmetric matrix>
$$
H=\begin{pmatrix}A&B/2\\B/2&C\end{pmatrix},\qquad \Delta_2=-4\det H.
$$
An <orthogonal matrix> $R$ changes this matrix to $R^{\mathsf T}HR$, whose <determinant> equals $\det H$. The <spectral theorem for real symmetric matrices> then removes the mixed term. Equal-sign, opposite-sign and one-zero <eigenvalues> give the three types after the remaining linear terms and offsets are considered.
The qualification matters. The equations $x^2+y^2=0$, $y^2=0$ and $xy=0$ describe a <point>, a repeated <straight line>, and two crossing <straight lines>, respectively. Their quadratic discriminants have the signs expected of ellipse, parabola and hyperbola types, but all three are <degenerate conics>. Also, $x^2+y^2=-1$ has an <empty set> of real solutions despite its negative quadratic discriminant.
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