Exploration of the Locus of Points Equidistant from a Fixed Point and a Circle

by Sandy Cederbaum

In Instructional Unit 10 , we explored the geometric definition of a parabola as the locus of points equidistant from a line called the directrix, and a point called the focus. We were able to make a GSP sketch that allowed us to visualize what would happen if we moved the focus F further away form, or closer to the directrix. In this investigation, we will use many of the same tactics, but we will replace our directrix with a circle. Open GSP and follow along by drawing your own sketch.

Let's start with our "new directrix" (a circle) and a point. Simply select the circle tool from the toolbar and click and drag a relatively small circle in the window. Then select the point tool and put a point in the window as well (Again, for convenience sake, put the point about an inch or two above the circle.).

Refer back to the construction in Unit 10 and see if you can figure out how to construct the locus of points that are equidistant from the circle and the focus F. An explanation of the construction is here if you need it.

Here is a GSP Sketch that you can play with if you had any trouble completing your sketch.

Play with your sketch now by moving point A around your circle. Move the focus. Now see what happens as you move point A. In order to get a better picture of what is going on, goto the Edit menu. Choose Preferences...Color, and put a check in the box Fade Traces Over Time. Select point L and goto the Display menu and choose Trace Intersection. Play with the sketch some more. Move F close to the circle and move A. Move F farther from the circle and move A. Do you know the name for the locus of points that you are seeing traced out? Are these hyperbolas? Try moving F inside the circle. What happens now? Select point L and remove the trace feature by deselecting it in the Display menu. Now let's look at the locus of points. Select point L and point A. Goto the Display menu and choose Locus. Now go back to your sketch and play with it again. Put F anywhere outside the circle. What geometric figure do you see? Now put F inside the circle. Are there any changes? What happens when F is the center of the circle? What happens when F falls on the circle?

Here are some sketches of what you should see.

In the study of conic sections, the following definitions are often given.

Circle: The locus of points equidistant from a fixed point (the center).

Ellipse: The locus of points (x,y) in the plane such that the sum of the distances from (x,y) to two fixed points (foci) is constant.

Hyperbola: The locus of points (x,y) in the plane such that the difference of the distances from (x,y) to two fixed points (foci) is constant.

Are these definitions in keeping with what we have explored above or does our construction offer alternate definitions of these conics? If so, what are these alternate definitions?