# Definition:Conic Section/Directrix

< Definition:Conic Section(Redirected from Definition:Directrix of Conic Section)

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## Definition

Let $K$ be a conic section specified in terms of:

- a given straight line $D$
- a given point $F$
- a given constant $e$

where $K$ is the locus of points $P$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F$ are related by the condition:

- $q = e p$

The line $D$ is known as the **directrix** of the conic section.

## Historical Note

The focus-directrix definition of a conic section was first documented by Pappus of Alexandria.

It appears in his *Collection*.

As he was scrupulous in documenting his sources, and he gives none for this construction, it can be supposed that it originated with him.

## Sources

- 1933: D.M.Y. Sommerville:
*Analytical Conics*(3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $1 \text a$. Focal properties - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.8$: Pappus (fourth century A.D.): Appendix: The Focus-Directrix-Eccentricity Definitions of the Conic Sections - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**directrix**:**1.**(plural**directrices**) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**directrix**:**1.**(plural**directrices**)