---
tag: 081YA6RCX6
kind: definition
name: Directrix
msc: [51N20]
defines: 0E5CXYNEPX
uses: [00HJCMC2CD, 0RBKKNH4GV, 0MP9BCF6DP, 01AAYNSBN6]
sources:
  - url: https://en.wikipedia.org/wiki/Conic_section
    locator: Eccentricity, focus and directrix
  - url: https://en.wikipedia.org/wiki/Hyperbola
    locator: By the directrix property
---

## Definition

Let $Q$ be a non-degenerate [[conic section|00HJCMC2CD]] in the Euclidean plane $\mathbb{R}^2$ that is not a circle. Let $F$ be a [[focus|0RBKKNH4GV]] of $Q$. For a point $P$ and a [[line|0MP9BCF6DP]] $\ell$, let $d(P, \ell)$ be the [[distance from $P$ to $\ell$|01AAYNSBN6]].
A directrix of $Q$ for the focus $F$ is a line $\ell$ that does not contain $F$, such that for a real number $e > 0$:
$$Q = \{\, P \in \mathbb{R}^2 \;:\; d(P, F) = e \, d(P, \ell) \,\}.$$

## Remarks

Each focus of $Q$ has exactly one directrix, and the number $e$ is the [[eccentricity|0HVX863SAJ]] of $Q$. The conic is an ellipse if $e < 1$, a parabola if $e = 1$ and a hyperbola if $e > 1$.
For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ or the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the directrices are the lines $x = \pm a^2 / c$, for the foci $(\pm c, 0)$.
A circle has no directrix in the Euclidean plane.

## Explanations

### L1 human

For a parabola, the directrix is the line in its definition: each point is as far from the focus as from the directrix. For an ellipse, each point is closer to the focus than to the directrix, by a fixed factor $e < 1$. For a hyperbola, the factor is greater than $1$.