## Parabolas

We have previously seen how a parabola is defined in terms of parametric equations or alternatively in Cartesian form.

An alternative way to define a parabola is as a *locus* of points.

## Focus and directrix

The locus defining a parabola depends on a *focus* and a *directrix*.

The focus is a point. For a standard parabola, the focus is located on the x axis a distance `$a$`

from the origin, that is at the
point (a, 0).

`$a$`

is the constant in the parabola equation `$y^2 = 4 a x$`

The directrix is a line. For a standard parabola it is a line perpendicular to the x axis passing through (-a, 0), that is
the line `$x = -a$`

The *vertex* of the parabola is the turning point. It is always halfway between the focus and directrix, which is always at the
origin for a standard parabola.

## The locus

A parabola is the locus of all points which are an equal distance from the focus and directrix. The diagram below illustrates this:

```
$$
FP = PD
$$
```

Press the play button to run it. As the purple dot traces the curve, notice that the two purple lines are alway equal length.