Definition:Conic Section/Focus-Directrix Property

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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 $K$.


Directrix

The line $D$ is known as the directrix of the conic section.


Focus

The point $F$ is known as the focus of the conic section.


Eccentricity

The constant $\epsilon$ is known as the eccentricity of the conic section.


Varieties of Conic Section

Circle

Conic section of eccentricity $0$:


CircleFocusDirectrix.png

It is not possible to define the circle using the focus-directrix property.

This is because as the eccentricity $e$ tends to $0$, the distance $p$ from $P$ to $D$ tends to infinity.

Thus a circle can in a sense be considered to be a degenerate ellipse whose foci are at the same point, that is, the center of the circle.


Ellipse

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


EllipseFocusDirectrix.png


Let $D$ be a straight line.

Let $F$ be a point.

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


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:

$\epsilon \, p = q$


Then $K$ is an ellipse.


Parabola

Conic section of eccentricity $1$:


ParabolaFocusDirectrix.png


Let $D$ be a straight line.

Let $F$ be a point.


Let $K$ be the locus of points $P$ such that the distance $p$ from $P$ to $D$ equals the distance $q$ from $P$ to $F$:

$p = q$


Then $K$ is a parabola.


Hyperbola

Conic section of eccentricity greater than $1$:


HyperbolaFocusDirectrix.png


Let $D$ be a straight line.

Let $F_1$ be a point.

Let $\epsilon \in \R: \epsilon > 1$.


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:

$\epsilon \, p = q$


Then $K$ is a 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.


Sources