Exploration of Parabola in GSP

by Venessa Brown

A parabola can be defined as 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.

The goal of this assignment is to:

1. Construct a parabola given a fixed point for the focus and a line (segment) for the directrix.

2. Use an Action Button to generate the parabola from an animation and trace of a constructed point.

3. Repeat 9a with a trace of the tangent line at the constructed point.

**How to construct a parabola given a line segment and a fix point not on the line**

1. First construct a line (m) and a point (F) not on line. This line will be the directrix for our parbola and the point will be the focus.

2. Construct a point (A) on line m.

3. Then, create a segment (FA) from point F to point (A).

4. Construct a perpendicular line (o) at point (A) to line (m).

5. Also, construct a perpendicular bisector (n) of segment (FA). notice that the two perpendicular segments should intersect.

6. Construct point (B) at the intersection of line (o) and line (n).

7. Construct segment FB.

8. Construct Circle centered at B passing through F.

9. Create a trace or a locus of B as point (A) moves along line (m).

Animation for the constructed parabola:

Click animation button to Tracing point B in the applet

Locus at point B

3. Click animation for trace of the tangent line at the constructed point