1. Goal: Find the equation, vertices, and foci of an ellipse.
8.2
Ellipses

2. What you’ll learn about
Geometry of an Ellipse
… and why
Ellipses are the paths of planets and comets around the
Sun, or of moons around planets.

3. Elliptical Orbits Around the Sun

5. Hale-Bopp Comet (1997)

6. prolate

7. Washington D.C. (pg. 598)

8. Paris Subway

9. Lithotripter (pg. 598)

10. Key Points on the Focal Axis of an Ellipse
Note: The foci and vertices always lie on the major axis.

11. center: critical points: endpoints of major and minor axes length of horizontal axis: length of vertical axis: focal radius (distance from center to a focus):

13. Example: Fill in the blanks and graph.
1. center:__________ a= _____ b= ____ major:__________ minor:__________ foci:_____________

14. Example: Fill in the blanks and graph.
2. 𝑥 𝑦 2 36 =1 center:__________ a= _____ b= ____ major:__________ minor:__________ foci:_____________

15. Example: Fill in the blanks and graph.
3. 𝑥 𝑦−1 2 9 =1 center:__________ a= _____ b= ____ major:__________ minor:__________ foci:_____________

16. Example: Finding an Equation of an Ellipse
The center is (0, 0). The length of the major axis (on the x-axis) is 6, and the length of the minor axis is 2.
Equation:

17. Example: Finding an Equation of an Ellipse
The center is (-2, 3). The length of the major axis (parallel to the x-axis) is 14, and the length of the minor axis is 6.
Equation:

18. Example: Finding an Equation of an Ellipse

19. Example: Find the Vertices and Foci of the Ellipse

20. Example: Find the Equation in Standard Form and Graph
x2 + 2x + 9y2 – 54y + 73 = 0