Warm-Up

1
Warm-Up
Write the standard equation of the circle with
the given radius and center.
1) 9 (0,0)
2) 1 (0,5)
3) 4 (-8,-1)
4) 5 (4,2)
2
10.4 Ellipses

• What are the major parts of an ellipse?
• What are the standard forms of an ellipse
  equation?
• How do you find the value of the foci c when
  given a b?
• How do you graph an ellipse?
• How do you find the equation of an ellipse from a
  graph?

3
Definition of Ellipse
An ellipse is the set of all points P in a plane
such that the sum of the distances from P to two
fixed points, F1 and F2, called the foci, is a
constant.
4
Standard Equation of an Ellipse
Horizontal Major Axis
a2 gt b2 a2 b2 c2
length of major axis 2a
length of minor axis 2b
5
Standard Equation of an Ellipse
Vertical Major Axis
a2 gt b2 a2 b2 c2
length of major axis 2a
length of minor axis 2b
6
Example 1
Write the standard equation for an ellipse with
foci at (-8,0) and (8,0) and with a major axis of
20. Sketch the graph.
length of major axis 2a
2a 20,
so a 10
a2 b2 c2
102 b2 82
b2 100 - 64
b2 36,
so b 6
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Example 2
Find the vertices and co-vertices of the ellipse.
vertices
(0,7) and (0,-7)
co-vertices
(4,0) and (-4,0)
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Example 3
Write the standard equation of the ellipse.
length of major axis 2a
2a 16,
so a 8
length of minor axis 2b
2b 8,
so b 4
9
Practice
Write the standard equation for an ellipse with
foci at (5,0) and (-5,0) and with vertices at
(9,0) and (-9,0). Sketch the graph.
10

• What are the major parts of an ellipse?
• Vertices, co-vertices, foci, center, major
  minor axis
• What are the standard forms of an ellipse
  equation?
• How do you find the value of the foci c when
  given a b?
• c2 a2 b2
• How do you graph an ellipse?
• Plot the vertices, the co-vertices and draw the
  ellipse.
• How do you find the equation of an ellipse from a
  graph?
• a vertex of major axis goes into the
  denominator under major axis letter, and b
  co-vertex of minor axis goes into the denominator
  under the minor axis letter in the formula.

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Assignment 10.4
p.612, 19-67 Skip 41, 47
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Warm-Up
Write the standard equation for an ellipse with
foci at (-5,0) and (5,0) and with a major axis of
18. Sketch the graph.
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10.4 Day 2 Ellipses

• What is the standard equation for an ellipse if
  the vertex has been translated?

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Standard Equation of a Translated Ellipse
Horizontal Major Axis
a2 gt b2 a2 b2 c2
length of major axis 2alength of minor axis 2b
15
Standard Equation of a Translated Ellipse
Vertical Major Axis
a2 gt b2 a2 b2 c2
length of major axis 2alength of minor axis 2b
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Example 1
An ellipse is defined by the equation 4x2 9y2
16x 18y 11. Write the standard equation and
identify the coordinates of the center, vertices,
co-vertices, and foci. Sketch the graph of the
ellipse.
4x2 16x 9y2 18y 11
4(x2 4x) 9(y2 2y) 11
4(x2 4x 4) 9(y2 2y 1) 11 4(4) 9(1)
4(x 2)2 9(y 1)2 36
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Example 1
Write an equation of the ellipse with foci at
(3,5) and (3,-1) and vertices at (3,6) and (3,-2).
Plot the given points and make a rough sketch.
The ellipse has a vertical axis, so its equation
is in the form of
Find the center which is halfway between the
vertices.
Find a and c a 6-24 c 5-23
Find b using b2 a2-c2
18

• What is the standard equation for an ellipse if
  the vertex has been translated?

19
Assignment 10.4 day 2
p. 612, 24-60 even p. 628, 17, 18, 24, 25