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.

 Definition:  An ellipse is the set of points such that the sum of the 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.

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 DE.  These axes are the two “segments of symmetry”.

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

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

Hint: Place point P on point E to establish any relationships.  Also, place point P on point C.  What do you notice?

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

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

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

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

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

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 a^2 and b^2 positions are interchanged when the ellipse is stretched in the y-direction.  The distance from the center to each vertex  is a units  and the distance from the center to the endpoints of the minor axis is b units.  Recall from the previous exercise that an ellipse’s two foci are located c units from the center on the major axis and are found by applying the relationship a^2 – b^2 =c^2.  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   .

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 (h,k) is   .     The radius is r units.

Students should use Graphing Calculator to plug in different values for (h,k) and a and b for the ellipse.  With each different ellipse they create they should record the location of the foci and the length of the latus rectum.  They should make notes of any conclusions about the effects of changing the values.  See attached worksheet obtained from Virginia Laird’s Lesson Plan 3 on Conic Sections.