# Definition:Parabola

## Contents

## Definition

### Intersection with Cone

Let $C$ be a double napped right circular cone whose base is $B$.

Let $\theta$ be half the opening angle of $C$.

That is, let $\theta$ be the angle between the axis of $C$ and a generatrix of $C$.

Let a plane $D$ intersect $C$.

Let $\phi$ be the inclination of $D$ to the axis of $C$.

Let $K$ be the set of points which forms the intersection of $C$ with $D$.

Then $K$ is a **conic section**, whose nature depends on $\phi$.

Let $\phi = \theta$.

Then $K$ is a parabola.

### Focus-Directrix Property

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**.

## Parts of Parabola

Consider a parabola $P$ whose focus is at $F$ and whose directrix is $D$.

### Vertex

The **vertex** of $P$ is the point where the axis intersects $P$.

### Axis

The **axis** of $P$ is the straight line passing through the focus of $P$ perpendicular to the directrix $D$.

### Latus Rectum

The **latus rectum** of $P$ is the chord of $P$ passing through the focus of $P$ parallel to the directrix $D$.

## Also see

- Results about
**parabolas**can be found here.

## Historical Note

Some sources suggest that the word **parabola** was provided by Apollonius of Perga, who did considerable work on establishing its properties.

However, it is also believed that Menaechmus may have used the term, and that it may go back even further than that.

## Linguistic Note

The word **parabola** is pronounced with the stress on the second syllable: **par- a-bo-la**.

The plural of **parabola** is properly **parabolae**, but this is considered pedantic, and the usual plural form found is **parabolas**.

The form **parabolas** is used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**parabola** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**parabola** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**parabola**