1. Conics Ellipses

2. Ellipse Definition: the set of all points in a plane whose distances from two fixed points in the plane have a constant sum. The fixed points are the foci of the ellipse. The line through the foci is called the focal axis. The point on the focal axis are the vertices of the ellipse.

5. To find c use the Pythagorean relation: a²=b²+c²

6. Ellipse with center (0,0) HorizontalVertical Focal Axisx-axis y=0 y-axis x=0 Foci (±c,0)(0, ±c) Vertices (±a,0)(0, ±a) Semi major axis a a Semi minor axis b b To find c use the Pythagorean relation: a²=b²+c²

7. E1 Graph the ellipse. Label the center and 4 other points. Then list the vertices and foci

8. E1

9. E2 Graph the ellipse. Label the center and 4 other points. Then list the vertices and foci 64x² + 4y² = 256

10. E2

11. Ellipse with center (h,k)

12. Ellipse with center (0,0) HorizontalVertical Focal Axis y=k x=h Foci (h±c,k)(h, k±c) Vertices (h±a,k)(h, k±a) Semi major axis a a Semi minor axis b b To find c use the Pythagorean relation: a²=b²+c²

13. E3 Graph the ellipse. Label the center and 4 other points. Then list the vertices and foci

14. E3

15. E4 Graph the ellipse. Label the center and 4 other points. Then list the vertices and foci

16. E4

17. E5 Graph the ellipse. Label the center and 4 other points. Then list the vertices and foci

18. E5

19. Eccentricity The eccentricity is the ratio of c to a e=c/a