# 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 $K$.

### 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 $0$:

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 $1$:

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 $1$:

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 $1$:

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

- 1933: D.M.Y. Sommerville:
*Analytical Conics*(3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $1 \text a$. Focal properties - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.8$: Pappus (fourth century A.D.)