Equations of Ellipses with Centers at the Origin 5807

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Equations of Ellipses with Centers at the Origin
5/8/07
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Write an Equation for a Graph

• Write an equation for the ellipse shown.
• First, find the values of a and
• B for the ellipse. The length of
• the major axis is the distance
• between (0 , 6) and (0 , -6).
• 2a 12
• a 6
• The foci are located at (0 , 3) and (0 , -3), so
  c 3. Can use the relationship between a, b,
  and c to determine the value of b.
• c2 a2 - b2 (notice that a is the
  hypotenuse, NOT c).
• 9 36 b2
• -27 -b2
• b2 27

(0 , 6)
y
(0,3)
O
x
(0,-3)
(0 , -6)
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Write an Equation for a Graph Cont.

• Since the major axis is vertical, substitute 36
  for a2 and 27 for b2 in the form
• An equation for this ellipse is

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Write an Equation Given the Lengths of the Axes

• Write an equation of the ellipse. Assume that
  the center is at the origin and the major axis is
  horizontal.
• The length of the major axis is
• The length of the minor axis is

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Write an Equation given the lengths of the Axes
Cont.

• Major axis Minor axis
• Substitute a and b into the form
• An equation for the ellipse is

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Equations of Ellipses with Centers at (h , k)
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Graph an Equation in Standard Form

• Find the coordinates of the center and foci and
  the lengths of the major and minor axes of the
  ellipse with equation
  Then graph the ellipse.
• The center of this ellipse is at (0 , 0).
• Since Since
• The length of the major axis is 2(4) 8 units.
• The length of the minor axis is 2(2) 4 units.
• Since the x2 term has the greater denominator,
  the major axis is horizontal.

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Graph an Equation in Standard Form (Cont.)

• c2 a2 b2 c2 42 22 c2 12
• The foci are at

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Graph an Equation Not in Standard Form

• Find the coordinates of the center and foci and
  the lengths of the major and minor axes of the
  ellipse with equation x2 4y2 4x 24y24 0.
  Then graph the ellipse.

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Graph an Equation Not in Standard Form (Cont.)

• The center is at (-2 , 3).
• The foci are at
• The length of the major axis is 8 units.
• The length of the minor axis is 4 units.
• Notice that this graph is translated 2 units to
  the left and up 3 units.

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More Practice!!!!

• Textbook p. 438 439 12 20 even, 28 36
  even.
• Homework Worksheet 8.3