by Courrey J. Alexander

A parabola is the set of
all points P in the plane that are equidistant from a fixed point F (focus) and
a fixed line d (directrix).

We know that the parabola has
following properties:

¯ The locus of points to the focus is equal to the
distance to the directrix.

(PF = PA)

¯
The coordinates of F,P,
and A are (a, b) for F, (x, y) for P and (x, -b) for A.

¯
Since PF = PA (x - a)^{2}+(y - b)^{2}
= (y + b)^{2}

¯
Here is the equation of
the parabola y =
(x - a)^{2}/4b.

The GSP construction
of the parabola is as follows:

1. Construct a line segment and a point not on the line.
This segment will be the directrix, d.
The point not on the line will be the point for the focus, F.

2. Choose a point on the directrix, A, and construct the
perpendicular line to d through A.

3. Then construct the line segment FA.

4. Construct the perpendicular bisector to FA. This
bisector will intersect the perpendicular line to d through A at point P.

The GSP construction of
the ellipse is as follows:

1. Construct a circle, d (similar to d in for the
directrix of the parabola), and a point, F, inside the circle (any point other
than the center of the circle).

2. Choose a point, A, on the circle.

3. Draw the line through A and the center of the circle.

4. Construct the line segment FA.

5. Construct the perpendicular bisector to FA. This
segment will intersect the line through the center at point P.

The GSP construction
of the hyperbola is the same as the construction for the ellipse. The difference for the hyperbola is for
step 1, the focus, F, moves outside of the circle.