# Definition:Conic Section/Focus

## Definition

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

- a given straight line $D$
- a given point $F$
- a given constant $\epsilon$

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:

- $q = \epsilon \, p$

The point $F$ is known as the **focus** of the conic section.

## Linguistic Note

The word **focus** is of Latin origin, hence its irregular plural form **foci**.

It was introduced into geometry by Johannes Kepler when he established his First Law of Planetary Motion. The word in Latin means **fireplace** or **hearth**, which is appropriate, considering the position of the sun.

The pronunciation of **foci** has a hard **c**, and is rendered approximately as ** folk-eye**.

Beware the solecism of pronouncing it ** fo-sigh**, which is incorrect.

## 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