# The Parabola

You can also make a parabola using the spreadsheet, see GeoGebra Tutorial - Spreadsheet.

The parabola is one of the conic sections, described by Apollonius in his book "Conics".

An ancient definition of a parabola is:

Given a line (**a directrix**) and a point (**the focus
point**) find a point `P` with the distances (as in the picture)
`d _{1}=d_{2}`.
The parabola is the set of all such points.

The point `P` lies on the perpendicular bisector. The perpendicular
bisector intersects the parabola at one single point, the point `P`.
It is hence a **tangent line** to the parabola.

No other points on the perpendicular bisector lie on the parabola.

### Exercise1

Place the focus on the `y`-axis at (0,`a`), the directrix
at the `x`-axis, and let the point `P` have the coordinates
(`x,y`). Find an expression for `y` in terms of `x`
by using the distances `d _{1}=d_{2}`.

(Apollonius could not do this exercise, he wrote his books about 1800 years before the Cartesian plane was invented.)

## The focus of a parabola

A ray falling onto a mirror will have a reflection such that:

the incident angle = the angle of reflection

### Exercise 2

Show that a ray falling onto a parabola perpendicular to the directrix will
be reflected to the focus point. In other words, show that `α=β`.

## Other Conics

By letting the directrix being a circle instead of a line, you can get the shapes of the the other conics. Draw a perpendicular bisector between the focus point and a point on the circle.

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License