Assignment 6:

"Parabola Construction"

by

Margo Gonterman

A **parabola** is the set of all points that are the same distance from a line, called the **directrix**, and a point, called the **focus**.

Parabola Construction

- Construct a line that will be the directrix

- Construct a point not on the line to be the focus

- Construct a free point P on the directrix

- Construct a line through P that is perpendicular to the directrix

- Connect P and the focus.

- Construct the perpendicular bisector of the line connecting P and the focus through the midpoint M.

- Mark the intersection of the perpendicular bisector and the line perpendicular to the directrix through P as X.

-X will trace the parabola as P moves along the directrix

- Line XM is the line tangent to the parabola

Proof that Construction Satisfies Definition

- Consider triangle PMX and triangle FMX.

- Segment XM is common to both triangles.

- Line XM is the perpendicular bisector of line PF.

Therefore:

PM=FM

Angle PMX=Angle FMX since they are both right angles

- By SAS (side-angle-side), the two triangle are congruent.

Specifically, segment PX=segment FX

- PX is the distance between X and the directrix.

- FX is the distance between X and the focus.

Therefore the point X is equidistant from the focus and the directrix.

**GSP Sketches**

Click HERE to see a GSP sketch of the construction of a parabola. The point X will trace as it is animated.

Click HERE to see a GSP sketch of the construction of a parabola. The tangent line will trace as it is animated.

Extensions

What happens when the directrix is not a line?

Click HERE to see what happens when the directrix is a circle and the focus is inside the circle.

Click HERE to see what happens when the directrix is a circle and the focus is outside the circle.