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10.4 Ellipses

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10.4 Ellipses By: L. Keali i Alicea An ellipse is a set of points such that the distance between that point and two fixed points called Foci remains constant The ... – PowerPoint PPT presentation

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Title: 10.4 Ellipses


1
10.4 Ellipses
  • By L. Kealii Alicea

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
v2
c
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
Equation Major Axis (length is 2a) Minor Axis (length is 2b) Vertices Co-Vertices

Horizontal Vertical (a,0) (-a,0) (0,b) (0,-b)

Vertical Horizontal (0,a) (0,-a) (b,0) (-b,0)
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) a7
  • The co-vertex is on the x-axis (-3,0) b3
  • The ellipse has a vertical major axis is of the
    form

10
Given the equation 9x2 16y2
144Identifyfoci, vertices, co-vertices
  • First put the equation in standard form

11
  • From this we know the major axis is horizontal
    a4, b3
  • 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 v7
  • So the foci are at (v7,0) (-v7,0)

12
Assignment
10.4 A (all) 10.4 B (1-18 even, 19-20)
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