Assignment 6 #9: Construct a Parabola

by

Laura Singletary

A parabola is the set of points equidistant from a line, called a 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.

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

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

c) Use the locus command to generate the parabola from a constructed point or the tangent line at that point.

Solution:

In order to begin, construct a line, call it the directrix, and a point not on the line, call it the focus. Also, construct a point on the line, label it x. Construct a segment with an endpoints: x and the focus. See below:

Find the midpoint of the segment between the focus and x. Call it m. Construct a perpendicular line to the segment through point m. Call this line p, where p is the perpendicular bisector of the segment. We must construct the perpendicular bisector p, because the perpendicular bisector of a line segment is the *locus* of all the points that are equidistant from its endpoints.

The next construction needed is a line perpendicular to the directrix through point x. Call this perpendicular line, line j. Construct the intersection of line j and line p. Call the intersection point: y.

Now, it is possible to create the parabola by moving the point X along the directrix, as X moves, so does the intersection point Y created by the perpendicular bisector of X and the Focus and the line perpendicular line to the directrix. The movement of point Y creates the parabola. Does this meet our understanding of what it means to be a parabola? What is a parabola? A *parabola* is the set of all points that are the same distance from a fixed line (the directrix) and a fixed point (the focus) not on the directrix.

A) Using an action button to generate the parabola from an animation and a trace of the constructed point:

B) Repeat Part A with a trace of the tangent line at the constructed point:

C) Use the locus command to generate the parabola from a constructed point or the tangent line at that point:

Want to consider some other explorations? Try this GSP file to continue your explorations, or try constructing it yourself :) What happens when you move the focus furthers from the directrix? Closer? What might happen if the point moving about the directrix wasn't on a line, but a circle (Look below!)?

Looks like an ellipse! Try the GSP file to explore other ideas.