**Lesson
3**

**Ellipses**

In this lesson students will become familiar with
the equations and graphs of ellipses.
The definition of an ellipse will be learned both algebraically and
using the distance relationship.
Students will learn how to construct an ellipse using Geometer’s
Sketchpad and how to prove that this construction is an ellipse. Applications of ellipses will be explored.

**Definition: **An ellipse is the set
of all points P in a plane such that the sum of the distances from P to two
fixed points, _{} and _{}, called the foci, is constant.

Some other important information includes the
following:

·
Major axis is represented by *2a* and is the longest axis of the
ellipse.

·
Minor axis is represented by 2b and is
the shortest axis of the ellipse.

·
Eccentricity is the measure of how round
or flat the ellipse is. It is the ratio
of the distance from the center to a focus, and the distance from the center to
a vertex. _{}.

·
The
center of the ellipse is represented by *(h,
k).*

·
The
relationship between *a, b, *and *c, *is *a ^{2}-b^{2}=c^{2}. *For an exploration on why, click here.

·
Standard
equations for the ellipse and the graph of each are shown below.

Note that when the center of the ellipse is
translated to a point *(h, k)* so that
the center is other than the origin, the equation will become

_{} for
figure 8, and _{} for figure 9.

Try constructing an ellipse using
GSP. First complete the wax paper
folding activity. Use the instructions provided to complete
this construction by clicking here. The ellipse that has been constructed is not
necessarily oriented to the standard coordinate system. After your construction is complete, prove
that the construction is an ellipse.
Show your proof using two different methods, geometric and algebraic. Click here
to explore the construction of an ellipse.

**Practice problem 1:** The moon orbits Earth in an elliptical path
with the center of the Earth at one focus.
The major axis of the orbit is 774,000 kilometers, and the minor axis is
773,000 kilometers.

1.
Using (0, 0) as the center of the
ellipse, write the standard equation for the orbit of the Moon around Earth.

2.
How far from the center of Earth is the
Moon at its closest point?

3.
How far from the center of Earth is the
Moon at its farthest point?

4.
Find the eccentricity of the Moon’s
orbit around Earth.

**Practice problem 2:** If the ellipse is defined by the equation *16x ^{2}+4y^{2}+96x+8y+84=0 *is
translated 4 units down and 7 units to the left, write the standard equation of
the resulting ellipse.