EMAT 6680 :: Write Up #6 :: Clay Kitchings

__Problem:__

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.

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.

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

Picture (GSP):

The red curve
(parabola) is formed by the traces of a constructed point. (Click the picture above to obtain your
own GSP file and view the animation on your GSP program.)

Next (part b), let
us trace the perpendicular bisector to the segment constructed from the Focus
to the arbitrary point B along the directrixÉ shall we?

Now we have traced
the perpendicular bisectors that form the tangent lines and thus define the
shape of the parabola.

Next, we are to
create a locus using the Locus command in GSP

Now, just for the
sake of curiosity and exploration, what would happen if we repeated the above ÒprocessÓ
but instead used the Parabola as our Òdirectrix?Ó

In order to
accomplish this, we need to construct a segment from C to the original Focus.
Then, our perpendicular line to the curve at any point is actually a
perpendicular line to the original tangent line (in blue). We shall investigate what happens with
the intersection of this perpendicular with the perpendicular bisector
constructed to the segment joining C and the Focus. (The picture has been
modified to fit in our screen.)

Using the Locus
Command from the Construct Menu (with Points B and C), we observe somewhat of a
ÒteardropÓ shape, which continues to the right and to the left.