Classification of conics by the discriminant

Statement

Let $Q = \{\, (x, y) \in \mathbb{R}^2 : A x^2 + B x y + C y^2 + D x + E y + F = 0 \,\}$ be a non-degenerate conic section. Let $\Delta = B^2 - 4AC$ be the discriminant. Then $Q$ is

• an ellipse if and only if $\Delta < 0$,
• a parabola if and only if $\Delta = 0$,
• a hyperbola if and only if $\Delta > 0$.

Moreover, $Q$ is a circle if and only if $A = C$ and $B = 0$.

Remarks

The discriminant is $-4$ times the determinant of $\begin{pmatrix} A & B/2 \\ B/2 & C \end{pmatrix}$, the matrix of the quadratic part. A rotation or a translation of the coordinates does not change $\Delta$ or $A + C$. So the type of a conic does not depend on the choice of Cartesian coordinates. A hyperbola with $A + C = 0$ is rectangular: its asymptotes are perpendicular.

Explanations

L1 human

The terms of degree $2$ decide the shape of a conic. For $x^2 + y^2 = 1$, $\Delta = -4 < 0$ and we get a circle, a special ellipse. For $y = x^2$, $\Delta = 0$ and we get a parabola. For $x y = 1$, $\Delta = 1 > 0$ and we get a hyperbola.