Definition:Ellipse/Focus-Directrix

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

• Equivalence of Definitions of Ellipse

• Ellipse has Two Foci

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