Section 9-4

1. Section 9-4 Ellipses

2. Objectives • I can write equations for ellipses • I can graph ellipses with certain properties • I can Complete the Square on equations to find Standard Format

3. Ellipse • An ellipse is a set of points in a plane such that the sum of the distances from the two foci is a constant. • Major axis length is 2a • Minor axis is 2b

4. Ellipse Axis

5. Basic Ellipse Construction Major Axis = 2a; Minor Axis = 2b, Foci are a distance of “c” from the center point.

6. Major Axis is horizontal Center at (h, k) Standard equation is: Major Axis is vertical Center at (h, k) Standard equation is: Standard Ellipse Equations

7. Equations

8. Graphing Ellipses • Get equation into standard format • Determine whether horizontal or vertical?? • Determine key numbers “a”, “b”, and “c” from the equation and using pyth thm. • Graph center (h, k) • Graph major axis (2a) • Graph minor axis (2b) • Sketch in ellipse shape, then plot foci points on the major axis (“c”)

9. Example 1

10. Example 2

11. Example 3

12. Example 4

13. Example 5

14. Example 1 • Find the foci and length of major and minor axes for the ellipse with this equation: • 16x2 + 4y2 = 144 (1st divide all terms by 144) Since 36 > 9, then a2 = 36 and b2 = 9 a = 6; b = 3 b2 = a2 – c2 b2 = 36 – 9 = 27, so b = 5.2 Major Axis = 2a = 12 units Minor Axis = 2b = 6 units Foci are at (0, 5.2) and (0, -5.2)

15. Completing the Square • Given the following equation, find the length of the major and minor axes, plus location of foci • x2 + 9y2 – 4x + 54y + 49 = 0 • (x2 – 4x) + (9y2 + 54y) = -49 • (x2 – 4x) + 9(y2 + 6y) = -49 • (x2 – 4x + 4) + 9(y2 + 6y +9) = -49 + 4 + 81 • (x – 2)2 + 9(y + 3)2 = 36 (Now divide by 36)

16. Example Continued a2 = 36 , so a = 6 b2 = 4, so b = 2 c2 = a2 – b2 = 36 – 4 = 32, so c = 5.7 Major Axis = 2a = 12 units Minor Axis = 2b = 4 units Foci at (-5.7, 0) and (5.7, 0)

17. Homework • Worksheet 10-6