Definition:Conic Section/Focus-Directrix Property

Definition
A conic section is a plane curve which can be specified in terms of:
 * a given straight line $D$ known as the directrix
 * a given point $F$ known as a focus
 * a given constant $\epsilon$ known as the eccentricity.

Let $K$ be the locus of points $b$ such that the distance $p$ from $b$ to $D$ and the distance $q$ from $b$ to $F$ are related by the condition:
 * $(1): \quad q = \epsilon \, p$

Then $K$ is a conic section.

Equation $(1)$ is known as the focus-directrix property of $K$.

Circle
Conic section of eccentricity $0$:

Ellipse
Conic section of eccentricity between $0$ and $1$:

Parabola
Conic section of eccentricity $1$:

Hyperbola
Conic section of eccentricity greater than $1$:

Also see

 * Equivalence of Definitions of Conic Section