10.4 Ellipses

1. Whatyou should learn: Goal 1 Graph and write equations of Ellipses. 10.4 Ellipses Goal 2 Identify the Vertices, co-vertices, and Foci of the ellipse. 10.4 Ellipses

2. An Ellipse is a set of points such that the distance between that point and two fixed points called Foci remains constant P d1 d2 f1 f2 d1 + d2 = constant 10.4 Ellipses

3. The line that goes through the Foci is the Major Axis. • The midpoint of that segment between the foci is the Center of the ellipse (c) • The intersection of the major axis and the ellipse itself results in two points, the Vertices (v) • The line that passes through the center and is perpendicular to the major axis is called the Minor Axis • The intersection of the minor axis and the ellipse results in two points known as co-vertices 10.4 Ellipses

4. Graphing and Writing Equations of Ellipses • • An ellipse is the set of all points P such that the sum of the distances between P and two distinct fixed points, called the foci, is a constant. P • The line through the foci intersects the ellipse at two points, the vertices. d1 d2 focus focus The line segment joining the vertices is the major axis, and its midpoint is the center of the ellipse. d1+d2 = constant The line perpendicular to the major axis at the center intersects the ellipse at two points called co-vertices. The line segment that joins these points is the minor axis of the ellipse. The two types of ellipses we will discuss are those with a horizontal major axis and those with a vertical major axis. 10.4 Ellipses

5. vertex: (0, a) • co-vertex: (0, b) • focus: (0, c) • co-vertex: (–b, 0) co-vertex: (b, 0) vertex: (–a, 0) vertex: (a, 0) • • • • • • focus: (–c, 0) focus: (c, 0) minor axis focus: (0, –c) • major axis minor axis • co-vertex: (0, –b) major axis • vertex: (0, –a) x 2 a 2 x 2 b 2 y 2 b 2 y 2 a 2 + = 1 + = 1 y y x x Ellipse with vertical major axis Ellipse with horizontalmajor axis 10.4 Ellipses

6. cv1 F2 F1 v1 c v2 cv2 10.4 Ellipses

7. Example of ellipse with vertical major axis 10.4 Ellipses

8. Example of ellipse with horizontal major axis 10.4 Ellipses

9. Standard Form for Elliptical Equations Note that a is the biggest number!!! 10.4 Ellipses

10. The foci lie on the major axis at the points: • (c,0) (-c,0) for horizontal major axis • (0,c) (0,-c) for vertical major axis • Where c2 = a2 – b2 10.4 Ellipses

11. WRITING EQUATIONS Write the equation of an ellipse with center (0,0) that has a vertex at (0,7) & co-vertex at (-3,0) • Since the vertex is on the y-axis (0,7) a = 7 • The co-vertex is on the x-axis (-3,0) b=3 • The ellipse has a vertical major axis & is of the form 10.4 Ellipses

12. IDENTIFYING PARTS Given the equation 9x2 + 16y2 = 144Identify: foci, vertices, & co-vertices • First put the equation in standard form: 10.4 Ellipses

13. From this we know the major axis is horizontal & a = 4, b = 3 • So the vertices are (4,0) & (-4,0) • the co-vertices are (0,3) & (0,-3) • To find the foci we use c2 = a2 – b2 • c2 = 16 – 9 • c = √7 • So the foci are at (√7,0) (-√7,0) 10.4 Ellipses

14. Reflection on the Section How can you tell from the equation of an ellipse whether the major axis is horizontal or vertical? Write the equation in standard form: if the larger denominator is under the x, it is horizontal. Under the y, it is vertical. assignment Page 612 # 19 – 67 odd 10.4 Ellipses