Assignment 6

Explorations with Geometer's Sketchpad

by

Wenjing Li

9. 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.

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 generated the parabola from a constructed point or the tangent line at that point.

Given a fixed point as the focus and a line as the directrix, we construct a parabola. By the definition, a parabola is the set of points that have equal distance from the fixed point focus and the line directrix. We pick any point on the directrix and draw a line through this point and perpendicular to the directrix. Then draw a segment pass the point we picked on the directrix and the focus, find the midpoint of this segment and draw a red line pass through the segment and perpendicular to the segment. This red line will intersect the previous dashed perpendicular line at a point. This point will has the same distance from the focus and the directrix, so this point is on the parabola. If we trace the constructed point, it will generate a parabola. The following is the picture we generate.

We start with any focus point and directrix line.

Then we choose any point on the directrix.

We construct a segment connecting the point A and the focus point.

Then we find the midpoint of this segment.

Now we construct a line perpendicular to the segment and passes through the midpoint of the segment.

We construct a line perpendicular to the directrix passing through the point A, it will intersect the red line at the point C. From our construction we know that the distance of CA is equal to the distance of CP. By the definition of parabola, the point C is on the parabola.

Now we let the point A moves along the directrix line and trace the point C, we will get a parabola.

Now we gererate the parabola with the trace of the tangent line at the constructed point. The red line is always tangent to the parabola.

We use the locus command to generated the parabola from a constructed point or the tangent line at that point.

If we move the point we picked on the directrix along the line directrix and trace all the red tangent lines that tangent to the parabola, we see that those tangent lines will fill up the part of the plane that includes the line directrix and outside the parabola. The following pink lines are the trace of tangent lines of the parabola that we get.

The following is an animation of how to construct a parabola. Please click the following bottom, then click the green point on the directrix and move it along the directrix, we will get a parabola which is the red curve.

please click here to show an animation