Construction of a Parabola

By: Lacy Gainey

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.

We are going to construct a parabola given a fixed point for the focus and a line for the directrix.

Lets begin by constructing a line and a fixed point. The line will be the directrix and the fixed point will be the focus.

We know a parabola is a set of points equidistant from the directrix and the focus. Lets start out by finding a single point, S, that satisfies these requirements. If an isosceles triangle is formed between a point on the directrix, the focus, and S, when the point on the directrix and the focus form the base of the triangle, S is equidistant from the directrix and the focus. Remember that an isosceles triangle is a triangle with two congruent sides.

To construct this isosceles triangle:

- Construct a segment between a point on the directrix and the focus. Label the point on the directrix d.
- Find the midpoint of the segment.

- Construct a line through the midpoint, perpendicular to the segment.
- Construct a line through d, perpendicular to the directrix.
- S is the point where the two perpendicular lines intersect.

- Construct segments from S to the focus and S to d.

In the GSP file, drag point d and observe what happens to the dashed segments. Notice how these segments are always congruent no matter where we move d.

Tracing point S and animating point d, forms a parabola. Open the GSP file to see this animation. This parabola is a set of points equidistant from the directrix and the focus. Point S is one of the points in that set.

Now, instead of tracing point S, trace the tangent line to the parabola. The tangent line is the line we constructed through the midpoint of the base of our isosceles triangle. Open the GSP file to view this animation and observe what happens.

We have seen how we can construct the parabola by either tracing point S or the tangent line and animating point d. We can also construct the parabola by using the locus command. A locus is defined as a set of points that satisfy a certain condition. In our case, we are looking for a set of points equidistant from the directrix and the focus. Earlier, we discussed how this set of points was illustrated by tracing point S and animating point d. So to use the locus command to construct the parabola, select point S and point d, then click the locus button.