# Definition:Hyperbola/Focus-Directrix

## Definition

Let $D$ be a straight line.

Let $F_1$ be a point.

Let $e \in \R: e > 1$.

Let $K$ be the locus of points $P$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F_1$ are related by the condition:

- $e p = q$

Then $K$ is a **hyperbola**.

### Directrix

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

### Focus

The point $F_1$ is known as a **focus** of the hyperbola.

The symmetrically-positioned point $F_2$ is also a **focus** of the hyperbola.

### Eccentricity

The constant $e$ is known as the **eccentricity** of the hyperbola.

## Also see

## 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 - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**hyperbola**