Lesson 7 Operating on a Shoestring Solidify Understanding

Learning Focus

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 $8^{″} \times 8^{″}$ , two tacks, $36 inches$ of string, and a pencil.

1.

Cut three pieces of string: a $10 -inch$ piece, a $12 -inch$ piece, and a $14 -inch$ piece. Tie the ends of each piece of string together, making three loops.

2.

Place a piece of paper on top of the cardboard.

3.

Place the two tacks $4 inches$ apart, wrap the string around the tacks, and then press the tacks down.

4.

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.

5.

With your pencil in the loop and the string pulled tight, move your pencil around the path that keeps the string tight.

6.

What shape is formed? What geometric features of the figure do you notice?

7.

Repeat the process again using the other strings. What is the effect of changing the length of the string?

8.

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.

9.

The distance from the point on the ellipse to each of the two foci is labeled $d 1$ and $d 2$ .

How does $d 1 + d 2$ in Figure 1 compare to $d 1 + d 2$ in Figure 2? (Figure 1 and Figure 2 are the same ellipse.)

10.

How does $d 1 + d 2$ compare to the length of the ellipse, measured from one end to the other along the $x$ -axis? Explain your answer with a diagram.

You have just constructed an ellipse based upon the definition: An ellipse is the set of all points $(x, y)$ in a plane which have the same total distance from two fixed points called the foci. Like circles and parabolas, ellipses also have equations. The basic equation of the ellipse is derived in a way that is similar to the equation of a parabola or a circle. Since it’s usually easier to start with a specific case and then generalize, we’ll start with this ellipse:

11.

Now, use the conclusions that you drew earlier to help you to write an equation. (We’ll help with a few prompts.)

a.

What is the sum of the distances from a point $(x, y)$ on this ellipse to $F 1$ and $F 2$ ?

b.

Write an expression in terms of $x$ and $y$ for the distance between the point $(x, y)$ on the ellipse and $F 1 (- 3, 0)$ .

c.

Write an expression in terms of $x$ and $y$ for the distance between $(x, y)$ on the ellipse and $F 2 (3, 0)$ .

d.

Use your answers to a, b, and c to write an equation.

12.

The equation of this ellipse in standard form is:

$\frac{x^{2}}{16} + \frac{y^{2}}{7} = 1$ .

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 $(4, 0)$ and $(0, - \sqrt{7})$ . Do both of these points make the equation you wrote in 11d true? Show how you checked them here.

13.

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.

$a$ = horizontal distance from the center to the ellipse.

$b$ = vertical distance from the center to the ellipse.

$c$ = distance from the center to a focus.

Based on the diagram, describe in words the following expressions:

$2 a$ :

$2 b$ :

$2 c$ :

14.

What is the mathematical relationship between $a$ , $b$ , and $c$ ?

15.

Now you can use the standard form of the equation of an ellipse centered at $(0, 0)$ , which is:

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ .

Write the equation of each of the ellipses pictured below:

a.

b.

c.

16.

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 $(0, - 5)$ and $(0, 5)$ and the major axis with a length of $14$ .

Takeaways

Ellipse:

Features of an ellipse:

Equation of an ellipse with center $(h, k)$ :

Vocabulary

ellipse
major axis, minor axis of an ellipse
Bold terms are new in this lesson.

Lesson Summary

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.

Retrieval

1.

The rectangle in figure $B$ is a translation of the rectangle in figure $A$ . Write the equations of the two diagonals of rectangle $A B C D$ in point-slope form. Then write the equations of the two diagonals of $A^{'} B^{'} C^{'} D^{'}$ .

2.

Use the graph to find the missing values.

a.

$f (\underset{―}{}) = 5$

b.

$g (\underset{―}{}) = - 1$

c.

$g (\underset{―}{}) = 0$

d.

$f (0) + g (0) =$