The Parabola

Move the point along the line! The blue lines are perpendicular bisectors to the two points.

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) d₁=d₂. 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.

Exercise

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 d₁=d₂.

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