1. Ellipses Part 1 Circle/Ellipse Quiz: March 9 Midterm: March 11

2. Definition:  An ellipse is the set of all points in a plane such that the sum of the distance from P to two fixed point (F 1, F 2 ) call foci is constant.

3. Ellipses can have a Horizontal Major Axis or a Vertical Major Axis: Major Axis  Is the longer axis of the ellipse  The endpoints of the major axis are called VERTICES.  The coordinates for your foci are on the major axis. Minor Axis  Is the shorter axis of the ellipse  The endpoints of the minor axis are called CO-VERTICES.

4. Horizontal Major Axis

5. Vertical Major Axis

6. Note:  When the bigger number is under the x-term, the major axis will be on the x-axis (or parallel to it if translated)  When the bigger number is under the y-term, the major axis will be on the y-axis (or parallel to it if translated).

7. Note  The length of the major axis is 2a  The length of the minor axis is 2b  The foci are always on the major axis  The following are always true for ellipses: a 2 > b 2 a 2 – b 2 = c 2

8. Note  “a” is your vertices coordinate  “b” is your co-vertices coordinate  “c” is your foci coordinate

9. Example 1:  Identify the vertices, co-vertices, and foci of the ellipse with equation 9x 2 + y 2 = 36

10. Example 2:  Write the standard form equation for an ellipse with foci of (0, -4) and (0, 4) and with minor axis of 6.

11. Example 3:  Write the standard form equation for an ellipse with foci of (-8,0) and (8,0) with major axis of 20.

12. You Try:  Identify the vertices, co-vertices, and foci of the ellipse with equation

13. You Try:  Write the standard equation for an ellipse with foci (-12, 0) and (12, 0) and with a major axis of 26.

14. Homework  M3 book (GREEN) P. 188 #1-15

15. Ellipses Part 2 Circle/Ellipse Quiz: March 9 Midterm: March 11

16. Translated Ellipses  Horizontal Major Axis:  Vertical Major Axis:

17. Example 3:  Write the standard equation for an ellipse with its center at (2,-4) and with a horizontal major axis of 10 and minor axis of 6. Identify vertices and co-vertices.  **Use the value of a to find the vertices, and the value of b to find the co-vertices

18. You Try:  Write the standard equation for an ellipse with its center at (-1,-2) and with a vertical major axis of 8 and minor axis of 4.  Give the vertices and co-vertices

19. Example 4:  An ellipse is defined by 4x 2 + y 2 + 24x – 4y + 36 = 0. Write the standard equation, and identify the coordinates of its center, vertices, co-vertices, and foci. Sketch the graph.

20. The Graph:

21. You Try:  An ellipse is defined by the equation 4x 2 + 25y 2 – 24x + 200y + 336=0. Write the standard equation and identify the coordinates of the center, vertices, co-vertices, and foci.

22. The Graph:

23. Homework:  Re-work all the Ellipse examples from class You NEED to practice, practice, practice  Ellipse WS