1. End of Lesson 3

2. Lesson 4 Contents Example 1Write an Equation for a Graph Example 2Write an Equation Given the Lengths of the Axes Example 3Graph an Equation in Standard Form Example 4Graph an Equation Not in Standard Form

3. Example 4-1a Write an equation for the ellipse shown. In order to write an equation for the ellipse, we need to find the values of a and b for the ellipse. We know that the length of the major axis of any ellipse is 2 a units. In this ellipse, the length of the major axis is the distance between (0, 5) and (0, –5). This distance is 10 units.

4. Example 4-1a Divide each side by 2. The foci are located at (0, 4) and (0, –4), so c = 4. We can use the relationship between a, b, and c to determine the value of b. Equation relating a, b, and c Solve for b 2. and

5. Example 4-1a Since the major axis is vertical, substitute 25 for a 2 and 9 for b 2 in the form Answer: An equation of the ellipse is

6. Example 4-1b Write an equation for the ellipse shown. Answer:

7. Example 4-2a Sound A listener is standing in an elliptical room 150 feet wide and 320 feet long. When a speaker stands at one focus and whispers, the best place for the listener to stand is at the other focus. Write an equation to model this ellipse, assuming the major axis is horizontal and the center is at the origin. The length of the major axis is 320 feet. Divide each side by 2.

8. Example 4-2a The length of the minor axis is 150 feet. Divide each side by 2. Substitute and into the form Answer: An equation for the ellipse is

9. Example 4-2a How far apart should the speaker and the listener be in this room? The two people should stand at the two foci of the ellipse. The distance between the foci is 2c units. Equation relating a, b, and c Multiply each side by 2. Use a calculator. Substitute and Take the square root of each side.

10. Example 4-2a Answer: The two people should be about 282.7 feet apart.

11. Example 4-2b Sound A listener is standing in an elliptical room 60 feet wide and 120 feet long. When a speaker stands at one focus and whispers, the best place for the listener to stand is at the other focus. a. Write an equation to model this ellipse, assuming the major axis is horizontal and the center is at the origin. b.How far apart should the speaker and the listener be in this room? Answer: 103.9 feet apart Answer:

12. Example 4-3a Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation Then graph the equation. The center of this ellipse is at (0, 0). The length of the major axis is 2(6) or 12 units, and the length of the minor axis is 2(3 ) or 6. Since the x 2 term has the greatest denominator, the major axis is horizontal. Since and since

13. Example 4-3a Equation relating a, b, and c Take the square root of each side. The foci are at and

14. Example 4-3a You can use a calculator to find some approximate nonnegative values for x and y that satisfy the equation. xy 03 12.96 22.83 32.60 42.24 51.66 60 Since the ellipse is centered at the origin, it is symmetric about the y -axis. So, the points at (1, 2.96) and (–1, 2.96) lie on the graph. The ellipse is also symmetric about the x -axis, so the points at (1, –2.96) and (–1, – 2.96) also lie on the graph.

15. Example 4-3a Graph the intercepts (–6, 0) (6, 0) (0, 3) and (0, –3) and draw the ellipse that passes through them and the other points. Answer: center: (0, 0) ; foci: major axis: 12 ; minor axis: 6

16. Example 4-3b Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation Then graph the equation. center: (0, 0) ; foci: major axis: 10 ; minor axis: 4 Answer:

17. Example 4-4a Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation Then graph the ellipse. Complete the square to write in standard form. Original equation Complete the squares.

18. Example 4-4a Write the trinomials as perfect squares. Divide each side by 36.

19. Example 4-4a Answer:The center is (3, 2) and the foci are located at and The length of the major axis is 12 units and the length of the minor axis is 6.

20. Example 4-4b Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation Then graph the ellipse. center: (–2, 3) ; foci: major axis: 10 ; minor axis: 4 Answer: