10-3 Ellipses
TEKS 2A.5.B Algebra and geometry: sketch graphs of conic sections to relate simple
parameter changes in the equation to corresponding changes in the graph.
Who uses this?
The whispering gallery at the Chicago
Museum of Science and Industry
was designed by using an ellipse.
(See Exercise 31.)
Objectives
Write the standard
equation for an ellipse.
Graph an ellipse, and
identify its center,
vertices, co-vertices, and
foci.
Vocabulary
ellipse
focus of an ellipse
major axis
vertices of an ellipse
minor axis
co-vertices of an ellipse
Also 2A.2.A, 2A.5.C,
2A.5.D
If you pulled the center of a circle
apart into two points, it would stretch
the circle into an ellipse.
An ellipse is the set of points P(x, y)
in a plane such that the sum of the
distances from any point P on the
ellipse to two fixed points F 1 and
F 2, called the foci (singular: focus),
is the constant sum d = PF 1 + PF 2.
This distance d can be represented
by the length of a piece of string
connecting two pushpins located
at the foci.
Point P
F1
F2
You can use the distance formula to
find the constant sum of an ellipse.
EXAMPLE
1
Using the Distance Formula to Find the Constant Sum of an Ellipse
Find the constant sum for an ellipse with foci F 1 (-3, 0) and
F 2 (3, 0) and the point on the ellipse (0, 4).
d = PF 1 + PF 2
Definition of the constant sum of an ellipse
d = √
(x 1 - x 3) 2 + (y 1 - y 3) 2 + √
(x 2 - x 3) 2 + (y 2 - y 3) 2
Distance Formula
(-3 - 0) 2 + (0 - 4) 2 + √
(3 - 0) 2 + (0 - 4) 2
d = √
Substitute.
d = √
25 + √
25
Simplify.
d = 10
The constant sum is 10.
1. Find the constant sum for an ellipse with foci F 1 (0, -8)
and F 2 (0, 8) and the point on the ellipse (0, 10).
Instead of a single radius, an ellipse has two axes. The longer axis of an
ellipse is the major axis and passes through both foci. The endpoints
of the major axis are the vertices of the ellipse . The shorter axis of
an ellipse is the minor axis . The endpoints of the minor axis are the
co-vertices of the ellipse . The major axis and minor axis are perpendicular
and intersect at the center of the ellipse.
736
Chapter 10 Conic Sections
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The standard form of an ellipse centered at (0, 0) depends on whether the major
axis is horizontal or vertical.
Horizontal
Vertical
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The values a, b, and c are related by the equation c 2 = a 2 - b 2. Also note that the
length of the major axis is 2a, the length of the minor axis is 2b, and a > b.
Center at (0, 0)
Standard Form for the Equation of an Ellipse
MAJOR AXIS
EXAMPLE
2
HORIZONTAL
VERTICAL
Equation
y2
x2 + _
_
=1
2
a
b2
y2
x2 = 1
_
+_
2
a
b2
Vertices
( a, 0), (-a, 0)
(0, a ), (0, -a )
Foci
( c, 0), (-c, 0)
(0, c ), (0, -c )
Co-vertices
(0, b ), (0, -b )
( b, 0), (-b, 0)
Using Standard Form to Write an Equation for an Ellipse
Write an equation in standard form for each ellipse with center (0, 0).
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Step 1 Choose the appropriate form of equation.
y2
x2 + _
_
= 1.
The horizontal axis is longer.
a2 b2
Step 2 Identify the values of a and c.
a = 10
The vertex (-10, 0) gives the value of a.
c=8
The focus (8, 0) gives the value of c.
Step 3 Use the relationship c 2 = a 2 - b 2 to find b 2.
8 2 = 10 2 - b 2
Substitute 10 for a and 8 for c.
b 2 = 36
Step 4 Write the equation.
y2
x2 + _
_
=1
Substitute the values into the equation of an ellipse.
100 36
10- 3 Ellipses
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Write an equation in standard form for each ellipse with center (0, 0).
B the ellipse with vertex (0, 8) and co-vertex (3, 0)
Step 1 Choose the appropriate form of equation.
2
y2 _
_
The vertex is on the y-axis.
+ x2 = 1
2
a
b
Step 2 Identify the values of a and b.
a=8
The vertex (0, 8) gives the value of a.
b=3
The co-vertex (0, 3) gives the value of b.
Step 3 Write the equation.
2
y2 _
_
Substitute the values into the equation of an ellipse.
+ x =1
64
9
Write an equation in standard form for each ellipse with
center (0, 0).
2a. Vertex (9, 0) and co-vertex (0, 5)
2b. Co-vertex (4, 0) focus (0, 3)
Ellipses may also be translated so that the center is not the origin.
Center at (h, k)
Standard Form for the Equation of an Ellipse
MAJOR AXIS
EXAMPLE
3
HORIZONTAL
VERTICAL
Equation
(y - k)
(x - h) 2 _
_
+
=1
2
a
b2
(y - k) _
(x - h) 2
_
+
=1
a2
b2
Vertices
(h + a, k), (h - a, k)
(h, k + a), (h, k - a)
Foci
(h + c, k), (h - c, k)
(h, k + c), (h, k - c)
Co-vertices
(h, k + b), (h, k - b)
(h + b, k), (h - b, k)
2
Graphing Ellipses
(x - 3) 2
_
Graph the ellipse
2
(y - 1)
+ _ = 1.
2
16
36
Step 1 Rewrite the equation as
(y - 1) 2
(x - 3) 2 _
_
+
= 1.
42
62
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Step 2 Identify the values of h, k, a, and b.
h = 3 and k = 1, so the center is (3, 1).
a = 6 and b = 4; Because 6 > 4, the major
axis is vertical.
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Step 3 The vertices are (3, 1 ± 6), or (3, 7) and
(3, -5), and the co-vertices are (3 ± 4, 1), or
(7, 1) and (-1, 1).
Graph each ellipse.
y2
x2 + _
3a. _
=1
64 25
738
(y - 4) 2
(x - 2) 2 _
_
3b.
+
=1
25
9
Chapter 10 Conic Sections
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EXAMPLE
4
Engineering Application
A road passes through a tunnel
in the form of a semi-ellipse.
In order to widen the road to
accommodate more traffic,
engineers must design a larger
tunnel that is twice as wide
and 1.5 times as tall as the
original tunnel. The design
for the original tunnel can
be modeled
by the equation
y2
x2
___
___
+ 64 = 1, measured in feet.
100
b
a
a
a. Find the dimensions of the larger tunnel.
Step 1 Find the dimensions of the original tunnel.
Because 100 > 64, the major axis of the tunnel is horizontal.
a 2 = 100, so a = 10 and the width of the tunnel is 2a = 20 ft.
b 2 = 64, so b = 8 and the height of the tunnel is 8 ft.
Step 2 Find the dimensions of the larger tunnel.
The width of the larger tunnel is 2(20) = 40 ft.
The height is 1.5(8) = 12 ft.
b. Write an equation for the design of the larger tunnel.
Step 1 Use the dimensions of the larger tunnel to find the
values of a and b.
For the larger tunnel, a = 20 and b = 12.
Step 2 Write the equation.
The equation in standard form for the larger
y2
y2
x2 + _
x2 + _
_
tunnel is _
=
1,
or
= 1.
400 144
20 2 12 2
x
Engineers have designed a tunnel with the equation ___
+ ___
= 1,
64
36
measured in feet. A design for a larger tunnel needs to be
twice as wide and 3 times as tall.
4a. Find the dimensions for the larger tunnel.
4b. Write an equation for the design of the larger tunnel.
2
y2
THINK AND DISCUSS
1. Explain where the foci are located in relation to the vertices.
2. Compare circles and ellipses by using lines of symmetry.
3. GET ORGANIZED Copy and complete the graphic organizer. Give an
equation for each type of ellipse.
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10- 3 Ellipses
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