*Circles and Ellipses*

This activity will explore the conics sections known as an
ellipses and circles. Students should be
given the following definition and then encouraged to explore a GSP model
representing the geometric definition.

distances from each point to two fixed points (the foci) is constant. |

*Please
see attached GSP file for a demonstration of the definition*

Students can notice changes that occur in the ellipse when the foci are moved around. What does the shape look like when the foci are moved farther apart? What does the shape look like when the foci are close together? What shape do you have when the foci are on top of each other? The purpose of this investigation is to help students realize that there are two degenerate cases of an ellipse: a circle and a straight line.

Click here to experiment with the different locations of the foci.

Through direct teaching explain to the students that every
ellipse has two important line segments.
One is called the major axis, line segment ** BC** shown below, and the
other is called the minor axis, line segment

The following questions that were obtained from ** Exploring
Conic Sections with The Geometer’s Sketchpad**, will lead students to
discover how to locate the focal points of a given ellipse and its
corresponding major and minor axes by

a. using a ruler and calculator

b. using a compass only

**Questions**

- Using
the attached GSP file that gives the measurement
of major axis
, determine the length of*BC*without taking any more measurements. Explain you reasoning.*F1P + F2P*

*Hint:*
Place point ** P **on point

- Using
the GSP file from problem number one, notice that point
lies at one end of the minor axis of the ellipse, and points*E*and*F1*are the ellipse’s focal points. The length of major axis*F2*is 11.39 cm. Without measuring determine the lengths of*BC*and*EF1*. Explain your reasoning.*EF2*

- Using
the attached GSP file , your answer from the
previous question along with the fact that the length of the minor axis
*(segment*is 8.54 to determine the lengths of segments**ED**)and*OF1*.*OF2*

- Based
on your answers from the previous questions, what is the relationship
between
, and*OC*,*OE**OF2?*

- Use the given measurements to calculate the focal points of the ellipse below. Explain your method.

- Explain how you would find the focal points using a compass only.

For answers to these questions click here

Once the students have experienced these physical properties, they should be introduced to the standard equation of an ellipse. The students should be given the following information,

The standard form of an ellipse that lies in the x-direction with center (h,k) is . The positions are interchanged when the ellipse is stretched
in the y-direction. The distance from
the center to each vertex is b^2
units and the distance from the center
to the endpoints of the minor axis is aunits. Recall from the previous exercise that an
ellipse’s two foci are located b units from the center on the
major axis and are found by applying the relationship c . The ellipse’s width is found by locating
the latus rectum, which is a set distance above and below each focus
determined by the value a^2 – b^2 =c^2. |

Recalling that a circle is a degenerate case of an ellipse, students should have the following.

A circle is the set
of all points in a plane equidistant from a fixed point, called the
center. The fixed distance is the
radius. The standard form of a circle
whose center is at the point . The radius is |

Students should use Graphing Calculator to plug in different
values for ** (h,k)** and