Conic section

Definition

Let $A, B, C, D, E, F$ be real numbers, with $A$, $B$ and $C$ not all zero. Let $$M = \begin{pmatrix} A & B/2 & D/2 \\ B/2 & C & E/2 \\ D/2 & E/2 & F \end{pmatrix}.$$ The set $Q = \{\, (x, y) \in \mathbb{R}^2 \;:\; A x^2 + B x y + C y^2 + D x + E y + F = 0 \,\}$ in Cartesian coordinates is a non-degenerate conic section if $\det M \neq 0$ and $Q$ is not empty.

Remarks

If $\det M = 0$, the set $Q$ is a degenerate conic: two lines, one line, one point or the empty set. Each non-degenerate conic is an ellipse, a parabola or a hyperbola. The discriminant $B^2 - 4AC$ tells which one. Equivalently, a non-degenerate conic is the intersection of a double cone with a plane that does not pass through the apex. Apollonius of Perga wrote a systematic work on conic sections, about 200 BC.

Explanations

L1 human

Cut a double ice cream cone with a flat knife. Depending on the tilt of the knife, the cut edge is a closed oval, a U shape or a curve with two branches. These are the ellipse, the parabola and the hyperbola, and they all have equations of degree $2$.