1. 11.3 Ellipses Objective: By the end of the lesson, you should be able to write an equation of an ellipse and sketch its graph.

2. Table of Contents Definition of an Ellipse Definition of Foci Definition of Vertices Definition of Major Axis Definition of Minor Axis Definition of Co-vertices Standard Equation of an Ellipse Vertical Major Axis Calculating the Coordinates of the Foci Eccentricity

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4. Vocabulary you need to know Ellipse – The set of all points (x,y) such that the sum of the distances between (x,y) and two distinct fixed points (foci) are constant. Cabri

5. Vocabulary you need to know Foci (plural of focus) - The two distinct points in an ellipse. Focus

6. Vocabulary you need to know Vertices – The two points at the intersection of the line through the foci and the ellipse. Cabri

7. Vocabulary you need to know Vertices – The two points at the intersection of the line through the foci and the ellipse. Vertex

8. Vocabulary you need to know Major Axis – The line segment joining the vertices Major Axis

9. Vocabulary you need to know Minor axis – The line segment perpendicular to the major axis at the midpoint. Minor Axis

10. Vocabulary you need to know Co-vertices – The endpoints of the minor axis Co-vertex

11. In a general ellipse with the center at the origin… a is the distance from the center to the vertex b is the distance from the center to the co-vertex a b Cabri

12. In a general ellipse… So the coordinate of the right vertex is (a,0). The coordinate of the left vertex is (-a,0). (-a,0) (a,0) a a 2a Cabri

13. In a general ellipse The coordinate of the top co-vertex is (0,b) The coordinate of the bottom vertex is (0,-b) (0,b) (0,-b) b b 2b

14. Standard equation of an ellipse with the center at the origin The standard form of the equation of a ellipse with center at (0,0) and major and minor axes of length 2a and 2b, where a>b, is as follows. Horizontal major axis Vertical major axis Cabri

15. Standard equation of an ellipse with the center at the origin Horizontal major axis Vertical major axis Notice the switch!!

16. What does an ellipse with a vertical major axis look like? Major axis Minor axis Vertex Co-vertex Focus

17. How do we calculate the coordinates of the foci? The foci of the ellipse lie on the major axis, c units from the center, where c 2 =a 2 -b 2 Cabri

18. How do we calculate the coordinates of the foci? The foci of the ellipse lie on the major axis, c units from the center, where c 2 =a 2 -b 2 c c (c,0) (-c,0)

19. What is Eccentricity? Eccentricity tells us how flat (or round) the ellipse is. Cabri

20. What is Eccentricity? Eccentricity tells us how flat (or round) the ellipse is. As e approaches 0, the ellipse becomes a circle. As e approaches 1, the ellipse flattens to a line.

21. Example 1 Write an equation of an ellipse whose vertices are (-5,0) and (5,0) and whose co-vertices are (0,-3) and (0,3). Find the foci of the ellipse.