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**Assignment 6**

**The Parabola as the Set of
Points Equidistant from a Line to a Fixed Point**

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A parabola can be defined
as the set of points equidistant from a given line, called the directrix, and a
fixed point, called the focus.
Let’s begin with the directrix, the focus, F, and variable point,
P.

A point that is
equidistant from the focus and P is on the perpendicular bisector of FP. Construct the perpendicular bisector of
FP.

Constructing the
perpendicular to the directrix through P will create an intersection, I, with
the perpendicular bisector FP.
Construct the segment IF.
An isosceles triangle is formed.
Thus, point I is equidistant from the focus and the directrix.

Tracing point I as point P
moves along the directrix creates the set of point equidistant from the
directrix to F. Click **HERE**
to view this animation.

As you can see, the set of
points equidistant from a line to a fixed point is a parabola.

In the picture above you can
see that the perpendicular bisector of FP is tangent to the parabola at point
I. Lets trace the perpendicular
bisector as P moves along the directrix.
Click **HERE** to view this animation.

Tracing the perpendicular
bisector of FP generates the envelope of the parabola.

You can also generate the
parabola using the locus command in GSP.
Highlight P and I. Under
the Construct menu click Locus.

So, the parabola
represents the set of points equidistant from a line to a fixed point. This can be represented using GSP as
done above.