Exploring the Geometric Definition of a Parabola

Sandy Cederbaum

A parabola is the set of points equidistant from a line, called the directrix, and a fixed point, called the focus. Assume the focus is not on the line. Construct a
parabola given a fixed point for the focus and a line (segment) for the directrix.

Here is a Geometer's Sketchpad script that shows the construction of the parabola. The first two points that you put in the plane represent the endpoints of the directrix. The third point represents the focus. If you move the point that appears on the directrix, you can trace out a parabola. Describe what happens when you move the focus away from the directrix. What happens when the focus is very close to the directrix? Make some conjecture about what will happen if the focus is placed on the directrix.

You can animate this parabola by clicking here and then double clicking on the animate button. When you want to stop the animation, simply click the mouse again.

Let us look at this animation again and trace the tangent line at the constructed point. Click here to see this animation.

Use the locus command to generated the parabola from a constructed point or the tangent line at that point. To construct the locus of an object, select that object, and then select some point constructed on a path which, when it moves along its path, defines the position of your selected object.

Here is the locus constructed from a point on our parabola.

Here is the locus constructed from the tangent line at that point.