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

• Equivalence of Definitions of Hyperbola

• Hyperbola has Two Foci

• Definition:Circle
• Definition:Ellipse
• Definition:Parabola

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