11'3 Ellipses

1
11.3 Ellipses

• An ellipse is the set of all points, (x,y), such
  that the sum of the distances between (x,y) and
  two distinct fixed points, the foci, is constant.
• Foci are two points that lie on the major axis of
  an ellipse.
• Eccentricity is the ovalness of an ellipse.
• The major axis is always a.
• The minor axis is always b.
• The foci is always c.

2
Ellipse on a Horizontal Axis
Co-vertex (0,b)
y
Center (0,0)
Vertex (-a,0)
Vertex (a,0)
x

• a gt b
• Foci lie on major axis c2 a2 - b2
• Eccentricity c/a
• x-axis is major axis 2a
• y-axis is minor axis 2b

Co-vertex (0,-b)
(c,0)
(-c,0)
3
Ellipse on a Vertical Axis
Vertex (0,a)
y
Center (0,0)
Co-vertex (b,0)
(0,c)
x

• a gt b
• Foci lie on major axis c2 a2 - b2
• Eccentricity c/a
• y-axis is major axis 2a
• x-axis is minor axis 2b

Co-vertex (-b,0)
Vertex (0,-a)
(0,c)
4
Example 1

• Write the equation for the ellipse given the
  following information
• vertices (-3,0) (3,0) co-vertices (0,-2)
  (0,2) center (0,0)
• Solution
• a gt b, 3 gt 2, therefore the vertices is on the
  x-axis, which is a horizontal ellipse.

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Example 1 Finding the Foci

• The foci of the ellipse are two points that lie
  on the major axis.
• In our example, the ellipse is horizontal, so the
  foci are on the x-axis.

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Example 2

• Write the equation and find the foci given
• vertices (0,-13) (0,13) co-vertices (-12,0)
  (12,0)
• Solution
• Remember a gt b, therefore a 13, b 12
• By observing the given information, the vertices
  is on the y-axis.
• Therefore the ellipse is vertical.

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Example 3

• Put the equation in standard form. Find a, b,
  and c.

• Note larger number is under x, therefore ellipse
  is horizontal.