10.4 Ellipses

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10.4 Ellipses

• By L. Kealii Alicea

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• 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
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cv1
F2
F1
v1
v2
c
cv2
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• 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

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Example of ellipse with vertical major axis
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Example of ellipse with horizontal major axis
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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!!!
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• 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

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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

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Given the equation 9x2 16y2
144Identifyfoci, vertices, co-vertices

• First put the equation in standard form

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• 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
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• c v7
• So the foci are at (v7,0) (-v7,0)

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