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.**

1. In order to begin construction, we must first make a line and a point of Focus. We should call the line, the “directrix line”. The point should be labeled as the focus.

2. Next, make a movable point on the directrix line. This point will help us to trace the
path of the parabola.

3. Now, we can construct a segment from the focus to the
movable point. We must also find a
point that is equidistant from the focus to the movable point. We can construct this point by finding
the midpoint of the segment.
Clicking the CONSTRUCT button at the top of the screen and choosing the
MIDPOINT option can create the midpoint.

4. The midpoint can now be used to construct a
perpendicular bisector through the segment from the movable point to the
focus. The perpendicular bisector
is representative of one of the tangent lines in our parabola.

5. By tracing the perpendicular bisector and animating
the movable point we are able to see our parabola being formed.

________________________________________________________

Our
construction above proves that all points on a parabola are equidistant from
the directrix and the focus.
Let’s look closer at finding a point on the parabola. By making a line of intersection with
our movable point, tangent line and directrix line, we can find a point on our
parabola.

__ __

__Other Explorations:__

** **

Let’s
investigate the construction of **ellipses**:

An ellipse is the set of all points whose sum of its distance from two foci is a constant.

1. We must construct a circle (directrix) and a point
(focus).

2. Now, make a movable point on the directrix circle by
highlighting the circle and construct a point on the circle.

Draw
a segment from the focus to the movable point on the circle and create the
perpendicular bisector of this segment.
The perpendicular bisector acts as one of the tangent lines in our
ellipse.

3. Next, in order to see all tangent lines of our
ellipse, we can

trace
the perpendicular bisector and animate the movable point.

View
the animation, by clicking here.

Notice
that the center and the focus are the foci of our ellipse.

_____________________________________________________

Let’s,
construct a **hyperbola**:

A hyperbola is the locus of all points, P, in the plane the difference of whose distances from two fixed points separated by a distance 2c is a given constant.

1. Similarly to our ellipse construction, we can
construct our

hyperbola
by creating a focus, circle and a movable point on the circle. Next, we can construct a segment that
joins the movable point and the focus.
Finding the midpoint of the constructed segment and creating a
perpendicular line through the midpoint and the constructed segment can make
the perpendicular bisector.

2. By animating the movable point and tracing the

perpendicular
bisector, we can see our hyperbola being developed.

Click
to see a complete animation of the hyperbola.

This
concludes our look at parabolas and other conic explorations.

___________________________________________________