# Constructing
Conic Sections

# by:
Lauren Wright

Conic sections are formed
by the intersection of parallel planes and a double cone - forming
ellipses, parabolas, and hyperbolas, respectively.

### PARABOLAS

A parabola can be defined
as the set of points equidistant from a line, called the directrix,
and a fixed point, called the focus. In other words, for any point
on a parabola, the length of the green line is equal to the length
of the blue line.
We are going to construct
a parabola using Geometer's Sketchpad. For our construction, we
will assume that the focus does not lie on the directrix line.
1. We will begin with
our directrix line and focus.
2. Next, we will make
a movable point on the directrix line to help us trace the path
of our parabola.
3. In order to find a
point that is equidistant from the movable point and the focus,
we will find the midpoint of the segment that connects them.
4. Using that midpoint,
we will draw a perpendicular bisector through the segment from
the movable point to the focus. This perpendicular bisector will
represent one of the tangent lines to the graph of the parabola.
5. With this one tangent
line, we can construct the set of all of the tangent lines to
the parabola by simply moving the movable point along the directrix
and tracing our tangent line.
6. With this set of tangent
lines, how can we find the points that actually lie on the parabola?
For that, we will have
to remember that every point on a parabola is equidistant from
the directrix and the focus. The distance from a point to a line
is measured by drawing the perpendicular through the point to
the line. So, if we draw a perpendicular through the movable point
to intersect our original tangent line, the intersection will
give us one point that lies on the parabola.
7. With this one point
on the parabola, we can construct the set of all of points by
simply moving the movable point along the directrix and tracing
our point on the parabola.
And, here is our parabola!
To investigate this construction yourself, click
here.

### ELLIPSES

An ellipse can be defined
as the set of all points whose sum of its distance from two foci
is a constant.
We are going to construct
an ellipse using Geometer's Sketchpad. This construction will
be much like the construction for the parabola. Only this time,
we will let a circle act as our "directrix." For this
construction, we will assume that the focus lies inside the circle.
1. We will begin with
our "directrix" circle and a focus.
2. Following the same
procedure as the parabola construction, we will construct a movable
point on the circle, a segment from the movable point to the focus,
and the perpendicular bisector of that segment.
3. Once again, this perpendicular
bisector is one of the tangent lines to the ellipse. We can construct
the set of all tangent lines by moving the movable point around
the circle and tracing the tangent line.
4. It is interesting
to note that our two foci turn out to be the original focus and
the center of the circle. Is this true for any focus inside the
circle? Let's find out.
5. It looks the two foci
will always be the original focus and the center of the circle.
Why does this ellipse appear to be more circular-shaped than our
previous one?
It's because the focus
we chose is closer to the center of the circle!
To investigate this construction
yourself, click here.

### HYPERBOLAS

A hyperbola can be defined
as the set of all points the difference of whose distances from
two foci is a constant.
We will construct a hyperbola
in Geometer's Sketchpad using the same methods we used for the
ellipse construction. The only difference will be that we choose
our focus to be outside the circle.
1. We will begin this
time with our "directrix" circle, our focus, and our
tangent line already constructed.
2. In the same manner
as before, we will trace the tangent line while moving the movable
point around the circle.
3. Here we can see that
our tangent lines form the shape of a hyperbola. Similar to the
ellipse, the two foci will be the original focus and the center
of the circle.
Ton investigate this
construction yourself, click
here.

#### This
concludes our investigations of conics!

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