Definition:Conic Section/Directrix

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)