---
tag: 0CEAT3418X
kind: definition
name: Conic section
msc: [51N20]
defines: 00HJCMC2CD
uses: [0RWKYX8339, 0XW7V9ZZ05]
sources:
  - url: https://en.wikipedia.org/wiki/Conic_section
    locator: General Cartesian form
  - url: https://en.wikipedia.org/wiki/Conic_section
    locator: Matrix notation
  - url: https://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections
    locator: Classification
---

## 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|0RWKYX8339]] 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|0VKCCEXBJ1]], a [[parabola|04S2GMHX82]] or a [[hyperbola|0WN5G117YC]]. The [[discriminant $B^2 - 4AC$|0N5NR32NQG]] 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 $2$.