Directrix

Builds on Conic section Focus of a conic Line Distance from a point to a line

Definition

Let $Q$ be a non-degenerate conic section in the Euclidean plane $\mathbb{R}^2$ that is not a circle. Let $F$ be a focus of $Q$ . For a point $P$ and a line $\ell$ , let $d(P, \ell)$ be the distance from $P$ to $\ell$ . 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 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.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Conic section concept
Euclidean distance concept
Focus of a conic concept
Line concept
Normal vector concept
Distance from a point to a line statement
Directrix definition

Tag	`081YA6RCX6`
URI	`m2://081YA6RCX6`
Kind	definition
Writing	W0 stub
MSC	51N20
Defines	Directrix
Uses	Conic section, Focus of a conic, Line, Distance from a point to a line

Directrix

Definition

Remarks

Reading path

Further reading

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