Lesson 7E Operating on a Shoestring Solidify Understanding
Understand the definition of an ellipse.
Understand relationships between parts of an ellipse.
Write the equation of an ellipse.
What shape is made by all the points that are a given distance from two points?
Open Up the Math: Launch, Explore, Discuss
You will need three pieces of paper, a piece of cardboard that is at least
Cut three pieces of string: a
Place a piece of paper on top of the cardboard.
Place the two tacks
Pull the string tightly between the two tacks, and hold them down between your finger and thumb. Pull the string tightly so that it forms a triangle, as shown below. What is the length of the part of the string that is not on the segment between the two tacks, the sum of the lengths of the segments marked
With your pencil in the loop and the string pulled tight, move your pencil around the path that keeps the string tight.
What shape is formed? What geometric features of the figure do you notice?
Repeat the process again using the other strings. What is the effect of changing the length of the string?
What is the effect of changing the distance between the two tacks? (You may have to experiment to find this answer.)
The geometric figure that you have created is called an ellipse. The two tacks each represent a focus point for the ellipse. (The plural of the word “focus” is “foci,” but “focuses” is also correct.) To “focus” our observations about the ellipse, we’re going to slow the process down and look at points on the ellipse in particular positions. To help make the labeling easier, we will place the ellipse on the coordinate plane.
The distance from the point on the ellipse to each of the two foci is labeled
You have just constructed an ellipse based upon the definition: An ellipse is the set of all points
Now, use the conclusions that you drew earlier to help you to write an equation. (We’ll help with a few prompts.)
What is the sum of the distances from a point
Write an expression in terms of
Write an expression in terms of
Use your answers to a, b, and c to write an equation.
The equation of this ellipse in standard form is:
It might be much trickier than you would imagine to re-arrange your equation to check it, so we’ll try a different strategy. This equation shows that the ellipse contains the points
Using the standard form of the equation is actually pretty easy, but you have to notice a few more relationships. Here’s another picture with some different parts labeled.
Based on the diagram, describe in words the following expressions:
What is the mathematical relationship between
Now you can use the standard form of the equation of an ellipse centered at
Write the equation of each of the ellipses pictured below:
Based on your experience with shifting circles and parabolas away from the origin, write an equation for this ellipse. Test your equation with some points on the ellipse that you can identify.
Ready for More?
Are you up for a challenge? Try finding the equation of the ellipse with foci at
Features of an ellipse:
Equation of an ellipse with center
In this lesson, we learned to understand the definition of an ellipse. We identified many of the features of an ellipse, including the foci, center, and major and minor axes. We found the equation of an ellipse based on the definition and learned to write the equation in standard form with any center.
The rectangle in figure
Use the graph to find the missing values.