Assignment 5 Write-Up

The Problem:

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 Part A 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.

The Strategy:

We need to construct the set of points (a parabola) which is equidistant from a given line (the directrix) and a given point (the focus). So we'll start with a line and put a point in any random place. What we'll need to do is create a point which is always equidistant from both the point and the line, then animate it or find the locus of those points.

Exploration:

Start with a line AB and a randomly-placed point not on the line, called C. AB will be the directrix and C will be the focus. In order to measure the distance from a point to a line, you need to take its perpendicular distance, so we'll need a random point on AB which we can animate in order to create our parabola. We'll call that point E. Construct a perpendicular to AB through point E. The point which will be used to create the parabola will be somewhere on this line. Now, draw a segment from C (our focus) to E (our random point on AB). Find the midpoint of segment CE and call it H. H will be the vertex of the parabola when E is directly below (or above) C. Now construct a perpendicular to CE through H. This line will always be tangent to the parabola, and where it intersects the line perpendicular to AB through E, is the tangent point. We'll call this point J. This is the point which we'll trace in order to create the parabola. (See the GSP sketches below for Parts A, B, and C. All of the sketches are completely dynamic, so you can move around line AB and point C to see what happens when you vary the distance between the directrix and the focus.)

GSP Sketch for Part A: Animation of traced point to create parabola

GSP Sketch for Part B: Animation of traced line tangent to parabola

GSP Sketch for Part C: Locus of points equidistant to a point and a line (a parabola)

Conclusions:

An easy extension to this problem would be to explore the locus of points which create other conic sections. You could explore the locus of points equidistant to a point (a circle) or the locus of points whose distance from two points adds up to a constant (an ellipse).