1. 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 and Vocabulary ellipse focus of an ellipse major axis minor axis vertices and co-vertices of an ellipse

2. Holt Algebra 2 10-3 Ellipses Notes 1. Find the constant sum for an ellipse with foci F 1 (2, 0), F 2 (–6, 0) and the point on the ellipse (2, 6 ). 2. Write an equation in standard form for each ellipse with the center at the origin. A. Vertex at (0, 5); co-vertex at (1, 0). B. Vertex at (5, 0); focus at (–2, 0). 3. Graph the ellipse. 4. A city park in the form of an ellipse is being renovated. The new park will have a length and width double that of the original park. Write the equation for the new park.

3. 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.

4. 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.

5. 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.

6. 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.

7. Holt Algebra 2 10-3 Ellipses The vertices are (±8, 0) or (8, 0) and (–8, 0), and the co-vertices are (0, ±5,), or (0, 5) and (0, –5). Example 2A : Graphing Ellipses Graph the ellipse

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

9. 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 2B: Graphing Ellipses Graph the ellipse

10. Holt Algebra 2 10-3 Ellipses Write an equation in standard form for each ellipse with center (0, 0). Example 3A: Using Standard Form to Write an Equation for an Ellipse Vertex at (6, 0); co-vertex at (0, 4) Step 1 Identify major axis - The vertex is on the x-axis. x2 x2 a2 a2 + = 1 y2 y2 b2 b2 x2 x2 36 + = 1 y2 y2 16 Substitute the values into the equation of an ellipse. Step 2

11. 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 3B: 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).

12. 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 3B Continued Substitute the values into the equation of an ellipse.

13. Holt Algebra 2 10-3 Ellipses Notes 16 1. Find the constant sum for an ellipse with foci F 1 (2, 0), F 2 (–6, 0) and the point on the ellipse (2, 6). 2. Write an equation in standard form for each ellipse with the center at the origin. A. Vertex at (0, 5); co-vertex at (1, 0). B. Vertex at (5, 0); focus at (–2, 0). 3. Graph the ellipse.

14. Holt Algebra 2 10-3 Ellipses 4. 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. Write the equation for the new park. Notes #4: Engineering Application x2 x2 49 + = 1 y2 y2 25

15. Holt Algebra 2 10-3 Ellipses Step 1: Find the dimensions of the existing park. Step 2: Find the dimensions of the new park. Step 3: Write the equation for the new park. x2 x2 49 + = 1 y2 y2 25 Length of major axis = 14 Length of minor axis = 10 Length of major axis = 28 Length of minor axis = 20 x2 x2 198 + = 1 y2 y2 100

16. Holt Algebra 2 10-3 Ellipses Reminder: Lesson Objectives Write the standard equation for an ellipse. Graph an ellipse, and identify its center, vertices, co- vertices, and foci. Ellipses: Extra Info The following power-point slides contain extra examples and information.

17. Holt Algebra 2 10-3 Ellipses Notes #3: Solutions 3. Graph the ellipse.

18. 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.

19. 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) Step 1 Choose the appropriate form of equation. The vertex is on the x-axis. x2 x2 a2 a2 + = 1 y2 y2 b2 b2 Check It Out! Example 2a

20. Holt Algebra 2 10-3 Ellipses Step 2 Identify the values of a and b. The vertex (9, 0) gives the value of a. x2 x2 81 + = 1 y2 y2 25 a = 9 b = 5 The co-vertex (0, 5) gives the value of b. Step 3 Write the equation. Substitute the values into the equation of an ellipse. Check It Out! Example 2a Continued

21. 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) Step 1 Choose the appropriate form of equation. The vertex is on the y-axis. y2 y2 a2 a2 + = 1 x2 x2 b2 b2 Check It Out! Example 2b

22. Holt Algebra 2 10-3 Ellipses Step 2 Identify the values of b and c. The co-vertex (4, 0) gives the value of b. y 2 25 + = 1 x2 x2 16 b = 4 c = 3 The focus (0, 3) gives the value of c. Step 3 Use the relationship c 2 = a 2 – b 2 to find a 2. Substitute 3 for c and 4 for b. Check It Out! Example 2b Continued Step 4 Write the equation. 3 2 = a 2 – 4 2 a 2 = 25

23. Holt Algebra 2 10-3 Ellipses Graph the ellipse. Step 1 Rewrite the equation as Step 2 Identify the values of h, k, a, and b. h = 2 and k = 4, so the center is (2, 4). a = 5 and b = 3; Because 5 > 3, the major axis is horizontal. Check It Out! Example 3b

24. Holt Algebra 2 10-3 Ellipses Step 3 The vertices are (2 ± 5, 4) or (7, 4) and (–3, 4), and the co- vertices are (2, 4 ± 3), or (2, 7) and (2, 1). Check It Out! Example 3b Continued (2, –1) (7, 4) Graph the ellipse.