Written by Sunny Yoon

By definition, a parabola is the set of points equidistant from a line, called the directrix, and a fixed point, called the focus. Assuming the focus is not on the line, here's the construction of a parabola given a fixed point for the focus and a line (segment) for the directrix.

1) Construct a line segment called the directix and a fixed point not on the directrix called the focus.

2) Construct a point on the directrix where you can move the point along the line segment.

3) Construct a perpendicular line to the directrix on the moving point.

4) Construct a segment using the moving point on the directrix and the focus.

5) Construct a midpoint from the above segment.

6) Construct a perpendicular line from the segment on the midpoint.

7) The intersection of two perpendicular lines is called the locus.

Here's a picture constructing a parabola locus using GSP.

Here's a picture where the locus is being traced.

When you trace the tangent line, here's the picture.

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

Here's a picture when I use the locus command from GSP.

Now, I'm going to repeat the steps of constructing a parabola locus, but this time, I'm going to use a circle instead of a line (segment) for the directrix.

What if the focus is inside the circle?

The locus became an ellipse!

What if the focus is outside the circle?

Then it creates a hyperbola locus.