1. Holt Algebra 2 10-3 Ellipses 10-3 Ellipses Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

2. Holt Algebra 2 10-3 Ellipses Warm Up If c 2 = a 2 – b 2, find c if c = ±12 1. a = 13, b = 5 2. a = 4, b = 3

3. Holt Algebra 2 10-3 Ellipses Write the standard equation for an ellipse. Graph an ellipse, and identify its center, vertices, co-vertices, and foci. Objectives

4. Holt Algebra 2 10-3 Ellipses ellipse focus of an ellipse major axis vertices of an ellipse minor axis co-vertices of an ellipse Vocabulary

5. Holt Algebra 2 10-3 Ellipses If you pulled the center of a circle apart into two points, it would stretch the circle into an ellipse. An ellipse is the set of points P(x, y) in a plane such that the sum of the distances from any point P on the ellipse to two fixed points F 1 and F 2, called the foci (singular: focus), is the constant sum d = PF 1 + PF 2. This distance d can be represented by the length of a piece of string connecting two pushpins located at the foci. You can use the distance formula to find the constant sum of an ellipse.

6. Holt Algebra 2 10-3 Ellipses Find the constant sum for an ellipse with foci F 1 (3, 0) and F 2 (24, 0) and the point on the ellipse (9, 8). Example 1: Using the Distance Formula to Find the Constant Sum of an Ellipse d = PF 1 + PF 2 Definition of the constant sum of an ellipse Distance Formula Substitute. Simplify. d = 27 The constant sum is 27.

7. Holt Algebra 2 10-3 Ellipses Find the constant sum for an ellipse with foci F 1 (0, –8) and F 2 (0, 8) and the point on the ellipse (0, 10). Check It Out! Example 1

8. Holt Algebra 2 10-3 Ellipses Instead of a single radius, an ellipse has two axes. The longer the axis of an ellipse is the major axis and passes through both foci. The endpoints of the major axis are the vertices of the ellipse. The shorter axis of an ellipse is the minor axis. The endpoints of the minor axis are the co- vertices of the ellipse. The major axis and minor axis are perpendicular and intersect at the center of the ellipse.

9. Holt Algebra 2 10-3 Ellipses The standard form of an ellipse centered at (0, 0) depends on whether the major axis is horizontal or vertical.

10. Holt Algebra 2 10-3 Ellipses The values a, b, and c are related by the equation c 2 = a 2 – b 2. Also note that the length of the major axis is 2a, the length of the minor axis is 2b, and a > b.

11. Holt Algebra 2 10-3 Ellipses Write an equation in standard form for each ellipse with center (0, 0). Example 2A: Using Standard Form to Write an Equation for an Ellipse Vertex at (6, 0); co-vertex at (0, 4) Step 1 Choose the appropriate form of equation. The vertex is on the x-axis. x2 x2 a2 a2 + = 1 y2 y2 b2 b2

12. Holt Algebra 2 10-3 Ellipses Example 2A Continued Step 2 Identify the values of a and b. The vertex (6, 0) gives the value of a. x2 x2 36 + = 1 y2 y2 16 a = 6 b = 4 The co-vertex (0, 4) gives the value of b. Step 3 Write the equation. Substitute the values into the equation of an ellipse.

13. Holt Algebra 2 10-3 Ellipses Co-vertex at (5, 0); focus at (0, 3) Step 1 Choose the appropriate form of equation. The vertex is on the y-axis. y2 y2 a2 a2 + = 1 x2 x2 b2 b2 Example 2B: Using Standard Form to Write an Equation for an Ellipse Step 2 Identify the values of b and c. The co-vertex (5, 0) gives the value of b. b = 5 c = 3 The focus (0, 3) gives the value of c. Write an equation in standard form for each ellipse with center (0, 0).

14. Holt Algebra 2 10-3 Ellipses y 2 34 + = 1 x2 x2 25 Step 3 Use the relationship c 2 = a 2 – b 2 to find a 2. Substitute 3 for c and 5 for b. Step 4 Write the equation. 3 2 = a 2 – 5 2 a 2 = 34 Example 2B Continued Substitute the values into the equation of an ellipse.

15. Holt Algebra 2 10-3 Ellipses Write an equation in standard form for each ellipse with center (0, 0). Vertex at (9, 0); co-vertex at (0, 5) Check It Out! Example 2a

16. Holt Algebra 2 10-3 Ellipses Write an equation in standard form for each ellipse with center (0, 0). Co-vertex at (4, 0); focus at (0, 3) Check It Out! Example 2b

17. Holt Algebra 2 10-3 Ellipses Ellipses may also be translated so that the center is not the origin.

18. Holt Algebra 2 10-3 Ellipses Example 3: Graphing Ellipses Step 1 Rewrite the equation as Step 2 Identify the values of h, k, a, and b. h = –4 and k = 3, so the center is (–4, 3). a = 7 and b = 4; Because 7 > 4, the major axis is horizontal. Graph the ellipse

19. Holt Algebra 2 10-3 Ellipses Step 3 The vertices are (–4 ± 7, 3) or (3, 3) and (–11, 3), and the co- vertices are (–4, 3 ± 4), or (–4, 7) and (–4, –1). Example 3 Continued Graph the ellipse

20. Holt Algebra 2 10-3 Ellipses Check It Out! Example 3a Graph the ellipse

21. Holt Algebra 2 10-3 Ellipses Graph the ellipse. Check It Out! Example 3b (7, 4)

22. Holt Algebra 2 10-3 Ellipses A city park in the form of an ellipse with equation, measured in meters, is being renovated. The new park will have a length and width double that of the original park. Example 4: Engineering Application x2 x2 50 + = 1 y2 y2 20

23. Holt Algebra 2 10-3 Ellipses Example 4 Continued Find the dimensions of the new park. Step 1 Find the dimensions of the original park. Because 50 > 20, the major axis of the park is horizontal. a 2 = 50, so and the length of the park is. b 2 = 20, so and the width of the park is

24. Holt Algebra 2 10-3 Ellipses Step 2 Find the dimensions of the new park. The width of the park is. The length of the park is. Example 4 Continued

25. Holt Algebra 2 10-3 Ellipses B. Write an equation for the design of the new park. For the new park,. The equation in standard form for the new park will be. Step 1 Use the dimensions of the new park to find the values of a and b. Step 2 Write the equation. Example 4 Continued

26. Holt Algebra 2 10-3 Ellipses Check It Out! Example 4 Engineers have designed a tunnel with the equation measured in feet. A design for a larger tunnel needs to be twice as wide and 3 times as tall. x2 x2 64 + = 1, y2 y2 36

27. Holt Algebra 2 10-3 Ellipses a 2 = 64, so a = 8 and the width of the tunnel is 2a = 16 ft. b 2 = 36, so b = 6 and the height of the tunnel is 6 ft. Check It Out! Example 4 Continued a. Find the dimensions for the larger tunnel. Step 1 Find the dimensions of the original tunnel. Because 64 > 36, the major axis of the tunnel is horizontal. Step 2 Find the dimensions of the larger tunnel. The height of the larger tunnel is 3(6) = 18 ft. The width of the larger tunnel is 2(16) = 32 ft.

28. Holt Algebra 2 10-3 Ellipses b. Write an equation for the larger tunnel. For the larger tunnel, a = 16 and b = 18. Check It Out! Example 4 Continued The equation in standard form for the larger tunnel is x2 x2 16 2 + = 1 or + = 1. y2 y2 18 2 x2 x2 256 y2 y2 324 Step 1 Use the dimensions of the larger tunnel to find the values of a and b. Step 2 Write the equation.