# The Parabola

## Why study the parabola?

The parabola has many applications in situations where:

- Radiation often needs to be concentrated at one point (e.g. radio telescopes, pay TV dishes, solar radiation collectors) or
- Radiation needs to be transmitted from a single point into a wide parallel beam (e.g. headlight reflectors).

Here is an animation showing how parallel radio waves are collected by a parabolic antenna. The parallel rays reflect off the antenna and meet at a point (the red dot, labelled F), called the **focus**.

## Definition of a Parabola

The **parabola** is defined as the **locus** of a point which moves so that it is always the same distance from a fixed point (called the **focus**) and a given line (called the **directrix**).The word **locus** means the set of points satisfying a given condition.

In the following graph,

- The
**focus** of the parabola is at (0,*p*). - The
**directrix** is the line *y *=−*p*. - The
**focal distance** is ∣*p*∣ (Distance from the origin to the focus, and from the origin to the directrix. We take absolute value because distance is positive.) - The
**point** (x, y) represents any point on the curve. - The
**distance** d from any point (x, y) to the focus (0,*p*) is the same as the distance from (x, y) to the directrix. - The
**axis of symmetry** of this parabola is the y-axis.

the** formula for a parabola**:

x^{2} = 4py

In more familiar form, with "y = " on the left, we can write this as:

y = x^{2}/ 4p

where p is the **focal distance** of the parabola.

A hyperbola is a pair of symmetrical open curves. It is what we get when we slice a pair of vertical joined cones with a vertical plane.

# The Hyperbola

### How do we create a hyperbola?

Take 2 fixed points A and B and let them be 4a units apart. Now, take half of that distance (i.e. 2a units).

Now, move along a curve such that from any point on the curve,

(distance to A) − (distance to B) =2*a* units.

The curve that results is called a **hyperbola**. There are two parts to the curve.