Lesson 2:

The Ellipse

By Carly Coffman

During this lesson we will explore the equation of an ellipse so that you will be able to recognize and manipulate it when you come in contact with it.

Open a Microsoft Word document, title it “Ellipse” and type your name and period under the title.   You will use this to answer any questions given in this lesson.  Remember to use complete sentences on all answers.  You do not have to type the questions.  Also, feel free to add comments and findings to the questions.

First, let’s look at an ellipse.

Terms:  An axis is the line of symmetry that runs vertical or horizontal in the ellipse.  The longer axis is called the major axis while the shorter axis is called the minor axis.

In this ellipse, if each tick mark represents one, the major axis has a length of 10 and the minor axis has a length of 6.

Now, we are going to manipulate some values in the equation to explore the equation of an ellipse.  Go back to the previous link and change the number “3” to a lower case “n”.  (If you do not see an equation put your cursor at the top edge of the graphing screen where it becomes two lines.  Then, click and drag down.  You should see a screen with an equation.)  Next, in the bottom left corner of the same graphing calculator screen click on n and make the window look like the following window.

Click on the play button on the bottom of the screen and watch what happens when n is changed.  On your Word document answer the following question.

1)              What variable is n connected with and how?

2)              How does changing n affect the ellipse?

Next, we will explore the constant connected with y.  So, use the following link to explore the constant, which will also be n.

3)                  Explain what the relationship is between this n and the

ellipse.

Well, this is a bit confusing!  We have two n’s.  Let’s give each one a different name so that you will know which n I am referring to later in the lesson. The first n we studied will be h.  The second n we studied will by k.

4)              Explain how to tell the difference between h and k when

looking at an equation for an ellipse.

Now, choose an h and k so that the ellipse is centered at the origin as shown below.

5)              Record what h and k have to be in order for the ellipse to be centered at the origin.  Also record the entire equation for this ellipse.  Now, double click on the picture above, copy it, and paste it into your Word document next to the equation you created.

Next, let’s look at manipulating the denominators of the equation. We will use an ellipse with a center of (0,0). Notice what happens as the denominator of the x-variable increases.

6)              Create a table of values with the denominator of the x-variable and the length

of the major axis.  What is the relationship?

Open the following link and click on the play button at the bottom to see how the ellipse changes when the denominator of the y-variable is manipulated.

7)              Create a table of values with the denominator of the y-variable and the length of the ellipse on the y-axis.  Use perfect squares for values of the denominator.  What is the relationship?

8)              What happens when you use a negative value for either denominator?  Is the

figure still an ellipse?  If not, what type of figure is it?

Now, let’s put our investigations together.  Copy and paste the following equation into your Word document.

Find each of the following for the equation above.

9)   Center:    ( __, ___ )

Length of the ellipse on the x-axis:

Length of the ellipse on the y-axis:

Foci

An ellipse is the set of all points (x,y) in a plane the sum of whose distances from two distinct fixed points (foci) is constant.  You can create your own ellipse by tying string to two thumbtacks, taking a pencil and stretching the string, and drawing around the thumbtacks so that the string is always tight.

The foci are found by using the equation, , where c is the distance from the center of the ellipse to each focus.  The coordinates of the foci are (h-c, k) and (h+c, k) if the major axis is horizontal and (h, k-c) and (h, k+c) if the major axis is vertical.

Let’s look at an example.

The center here is (4,-1).  The major axis is 10 units long, so a must be 5.  The minor axis is 6 units long, so b must be 3.  We can substitute these values into the equation,

The foci always lie on the major axis and are c units away from the center.  So, the foci are (4, -1+4) and (4, -1-4), which is (4, 3) and (4, -5).

In your word document add the foci for the equation in #9.

*Make sure your name is on your Word document and print it.  Place this document with your conic document in your portfolio or notebook.  You are finished with the ellipse lesson!