1. Sullivan Algebra and Trigonometry: Section 10.3 The Ellipse Objectives of this Section Find the Equation of an Ellipse Graph Ellipses Discuss the Equation of an Ellipse Work with Ellipses with Center at (h,k)

2. An ellipse is the collection of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant. F 1 : (-c,0)F 2 : (c,0) Minor Axis Major Axis Vertex

3. Equation of an Ellipse: Center at (0,0); Foci at (c, 0) and (-c,0); Major Axis is Horizontal where a > b > 0 and b 2 = a 2 - c 2 The major axis is the x - axis. The vertices are at (-a, 0) and (a, 0)

4. Find the equation of an ellipse with center at the origin, one focus at (4, 0), and a vertex at (-5,0). Graph the equation. Since the given focus and vertex are on the x- axis, the major axis is the x-axis. The distance from the center to one of the foci is c = 4. The distance from the center to one of the vertices is a = 5. Use c and a to solve for b. b 2 = a 2 - c 2 b 2 = 5 2 - 4 2 = 25 - 16 = 9 b = 3

5. So, the equation of the ellipse is: F 1 : (-4,0)F 2 : (4,0) (-5,0)(5,0) (0,3) (0,-3)

6. Discuss the equation: Since the equation is written in the desirable form, a 2 = 16 and b 2 = 7 Since b 2 = a 2 - c 2, it follows that c 2 = a 2 - b 2 or c 2 = 16 - 7 = 9. So, the foci are at (3,0) and (-3,0) The vertices are at (-4, 0) and (4, 0)

7. Equation of an Ellipse: Center at (0,0); Foci at (0, c) and (0, -c); Major Axis is Vertical where a > b > 0 and b 2 = a 2 - c 2 The major axis is the y - axis. The vertices are at (0, -a) and (0, a)

8. Find the equation of an ellipse having one focus at (0, 2) and vertices at (0, 3) and (0, -3). Since the vertices lie on the y - axis, the major axis is vertical with a = 3. The distance from the focus to the center is c = 2. b 2 = a 2 - c 2 b 2 = 3 2 - 2 2 = 9 - 4 = 5 So, the equation of the ellipse is:

9. If an ellipse with center at the origin and major axis coinciding with a coordinate axis is shifted horizontally h units and vertically k units, the resulting ellipse is centered at (h,k) and has the equation: Horizontal Major Axis Vertical Major Axis

10. Find the equation of an ellipse with center at (2, -3), one focus at (3, -3), and one vertex at (5, -3). The center is at (h,k) = (2, -3). So h = 2 and k = -3 The center, focus, and vertex all lie on the line y = -3, so the major axis is parallel to the x-axis and the ellipse is horizontal. The distance from the center to the vertex is a = 3. The distance from the center to the focus is c = 1. To solve for b,

11. b 2 = a 2 - c 2 b 2 = 3 2 - 1 2 = 9 - 1 = 8 So, the equation of the ellipse is: