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.

- 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!

Click
here for a GSP animation!

- 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!

Click
here for a GSP animation!

- 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!