# Definition:Ellipse/Focus-Directrix

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## Definition

Let $D$ be a straight line.

Let $F$ be a point.

Let $e \in \R: 0 < e < 1$.

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

- $e p = q$

Then $K$ is an **ellipse**.

### Directrix

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

### Focus

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

### Eccentricity

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

## 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