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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.


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


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.


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.