Write-Up #6

Parabola Construction

By Jaepil Han

9. Construct the locus of points equidistant from a fixed point F and a circle. In other words, repeat the parabola construction but use a circle as the "directrix." Let F be any point in the plane other than the center of the circle. Assume F is not on the circle; it can be either inside or outside.

First of all, construct a circle and a point F outside of the circle. Then, locate a point on the circle and make a segment between the constructed point and the point F. Now, plot a midpoint of the segment and construct a perpendicular line from the midpoint. Here's the picture of it.

Now, trace the perpendicular line from the midpoint of the segment. Here's the picture of it.

Or, we can simply use the "Locus" button and eventually obtain the following graph.

The graph is the locus of the perpendicular line when a point F located outside of the circle. As we can see, this constructs the hyperbola. The focal points are the point F and the center of the circle.

In addition, here's the locus of the perpendicular line when a point F located inside of the circle. To make it more dynamic, plotted the point F close to the circle.

The locus looks like an ellipse.

When we plot a point B on the curve and make segments BC and BF, the sum of the segments BC and BF is the same as the radius of the circle. By definition, the curve, the locus of the bisecting perpendicular line of the segment DF is an ellipse.

Here's an interesting thing. When the point F moves close to the center, the shape of the locus is getting more circular shape. For this reason, the shape might depend on the location of the point F, and more close to the center means more circular shape.