Generating the Parabola

By Colleen Garrett

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. Construct a parabola given a fixed point for the focus and a line (segment) for the directrix.

In this investigation I will

a. Use an Action Button to generate the parabola from an animation and trace of a constructed point.

b. Repeat 9a with a trace of the tangent line at the constructed point.

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

1. First, construct a line to be your directrix.  Construct a point Y on your directrix and a point off of the directrix to serve as the focus. Then construct a line segment between the focus and the point Y. Construct the perpendicular

bisector (b) of this segment. Draw a line (l) perpendicular to the directrix and through the point Y.  Construct a point

X where line l intersects line b. Trace this point and animate the point on your directrix.

You should get a parabola!

1. First, construct a line to be your directrix.  Construct a point Y on your directrix and a point off of the directrix to

serve as the focus. Then construct a line segment between the focus and the point Y. Construct the perpendicular

bisector (b) of this segment. Trace the perpendicular bisector and animate your point on your directrix.

You should get a parabola!

1. First, construct a line to be your directrix.  Construct a point Y on your directrix and a point off of the directrix to

serve as the focus. Then construct a line segment between the focus and the point Y. Construct the perpendicular

bisector (b) of this segment. Draw a line (l) perpendicular to the directrix and through the point Y.  Construct a point

X where line l intersects line b. Select points X and Y.  Use the construct tool to create a locus.  See our result

below!

We get a parabola again!