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10.4 Ellipses

10.4 Ellipses. p. 609. An ellipse is a set of points such that the distance between that point and two fixed points called Foci remains constant. d1. d2. f1. f2. d4. d3. d1 + d2 = d3 + d4. cv 1. F 2. F 1. v 1. c. v 2. cv 2. The line that goes through the Foci is the Major Axis.

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10.4 Ellipses

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  1. 10.4 Ellipses p. 609

  2. An ellipse is a set of points such that the distance between that point and two fixed points called Foci remains constant d1 d2 f1 f2 d4 d3 d1 + d2 = d3 + d4

  3. cv1 F2 F1 v1 c v2 cv2

  4. 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

  5. Example of ellipse with vertical major axis

  6. Example of ellipse with horizontal major axis

  7. Standard Form for Elliptical Equations Note that a is the biggest number!!!

  8. 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

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

  11. 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)

  12. Assignment

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