---
tag: 0CQW20XGFW
kind: definition
name: Parabola
msc: [51N20]
defines: 04S2GMHX82
uses: [0XW7V9ZZ05, 0NPPGHQNXQ, 0MP9BCF6DP, 01AAYNSBN6]
sources:
  - url: https://en.wikipedia.org/wiki/Parabola
    locator: Definition as a locus of points
  - url: https://en.wikipedia.org/wiki/Parabola
    locator: History
  - url: https://en.wikipedia.org/wiki/Conic_section
    locator: Standard forms in Cartesian coordinates
---

## Definition

Let $F$ be a point of the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$, and let $\ell$ be a [[line|0MP9BCF6DP]] that does not contain $F$. Let $d(P, F)$ be the [[Euclidean distance|0NPPGHQNXQ]], and let $d(P, \ell)$ be the [[distance from $P$ to $\ell$|01AAYNSBN6]].
The parabola with the focus $F$ and the directrix $\ell$ is the set
$$\{\, P \in \mathbb{R}^2 \;:\; d(P, F) = d(P, \ell) \,\}.$$

## Remarks

For $F = (a, 0)$ with $a > 0$ and $\ell$ the line $x = -a$, the parabola is the set of the points with $y^2 = 4 a x$.
For real numbers $p \neq 0$, $q$ and $r$, the graph of $y = p x^2 + q x + r$ is a parabola.
The [[eccentricity|0HVX863SAJ]] of each parabola is $1$. Any two parabolas are similar.
Menaechmus studied conic sections in the 4th century BC. Archimedes found the area of a parabolic segment, and the name is due to Apollonius.

## Explanations

### L1 human

A parabola is the set of points that are as far from a point as from a line. For the focus $(0, 1)$ and the line $y = -1$, the parabola is $y = x^2 / 4$.>>>