Chapter 0

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Section 9.2
The Hyperbola
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Objectives:
•
•
•
•
•
Locate a hyperbola’s vertices and foci.
Write equations of hyperbolas in standard form.
Graph hyperbolas centered at the origin.
Graph hyperbolas not centered at the origin.
Solve applied problems involving hyperbolas.
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Definition of a Hyperbola
A hyperbola is the set of
points in a plane the
difference of whose
distances from two fixed
points, called foci,
is constant.
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The Two Branches of a Hyperbola
The line through the foci
intersects the hyperbola at
two points, called the vertices.
The line segment that
joins the vertices is the
transverse axis.
The midpoint of the transverse
axis is the center of the hyperbola.
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Standard Forms of the Equations of a Hyperbola
The standard form of the equation of a hyperbola with
center at the origin is x 2 y 2
y 2 x2
 2  1 or 2  2  1.
2
a b
a b
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Standard Forms of the Equations of a Hyperbola
(continued)
The vertices are a units from the center and the foci are c
units from the center.
For both equations, b2 = c2 – a2. Equivalently, c2 = a2 + b2.
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Example: Finding Vertices and Foci from a Hyperbola’s
Equation
Find the vertices and locate the foci for the hyperbola with
the given equation: x 2 y 2
  1.
25 16
a 2  25, a  5. The vertices are (–5, 0) and (5, 0).
c2  a 2  b2
c 2  25  16  41
c   41


The foci are at  41,0 and


41,0 .
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Example: Finding Vertices and Foci from a Hyperbola’s
Equation (continued)
Find the vertices and locate the foci for the hyperbola with
the given equation:
2
2
x
y
 1
25 16
The vertices are (–5, 0) and (5, 0).
The foci are at
 41,0 and 41,0 .




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Example: Finding Vertices and Foci from a Hyperbola’s
Equation
Find the vertices and locate the foci for the hyperbola
2
2
y
x
with the given equation:
  1.
25 16
a  25, a  5. The vertices are (0, –5) and (0, 5).
2
c2  a 2  b2
c 2  25  16  41
c   41
The foci are at 0,  41 and 0, 41 .




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Example: Finding the Equation of a Hyperbola from Its
Foci and Vertices (continued)
Find the vertices and locate the foci for the hypberbola
with the given equation:
2
2
y x
 1
25 16
The vertices are (0, –5) and (0, 5).
The foci are at
0,  41 and 0, 41 .




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Example: Finding the Equation of a Hyperbola from Its
Foci and Vertices
Find the standard form of the equation of a hyperbola
with foci at (0, –5) and (0, 5) and vertices (0, –3) and
(0, 3).
Because the foci are located at (0, –5) and (0, 5), the
transverse axis lies on the y-axis. Thus, the form of the
equation is y 2 x 2
 2  1.
2
a b
The distance from the center to either vertex is 3, so
a = 3.
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Example: Finding the Equation of a Hyperbola from Its
Foci and Vertices
Find the standard form of the equation of a hyperbola
with foci at (0, –5) and (0, 5) and vertices (0, –3) and
(0, 3).
The distance from the center to either focus is 5. Thus,
c = 5.
The equation is
c2  a 2  b2
y 2 x2
 1
2
25  9  b
9 16
b 2  16
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The Asymptotes of a Hyperbola Centered at the Origin
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The Asymptotes of a Hyperbola Centered at the Origin
(continued)
As x and y get larger, the two branches of the graph of a hyperbola
approach a pair of intersecting straight lines, called asymptotes.
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Graphing Hyperbolas Centered at the Origin
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Example: Graphing a Hyperbola
2
2
x
y
Graph and locate the foci:
  1.
36 9
What are the equations of the asymptotes?
Step 1 Locate the vertices.
x2 y 2
The given equation is in the form 2  2  1
a b
2
2
with a = 36 and b = 9.
a2 = 36, a =  6. Thus, the vertices are (–6, 0) and (6, 0).
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Example: Graphing a Hyperbola (continued)
2
2
x
y
Graph and locate the foci:
  1.
36 9
Step 2 Draw a rectangle
a2 = 36, a =  6.
b2 = 9, b =  3.
The rectangle passes
through the points
(–6, 0) and (6, 0)
and (0, –3) and (0, 3).
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Example: Graphing a Hyperbola (continued)
2
2
x
y
Graph and locate the foci:
  1.
36 9
Step 3 Draw extended
diagonals for the rectangle
to obtain the asymptotes.
b 3 1
 
a 6 2
The equations for the asymptotes
1
1
are y  x and y   x.
2
2
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Example: Graphing a Hyperbola (continued)
2
2
x
y
Graph and locate the foci:
  1.
36 9
Step 4 Draw the two branches
of the hyperbola starting at
each vertex and approaching
the asymptotes.
c 2  a 2  b 2 c 2  36  9  45
c   45  6.4
The foci are at  45,0 and
or (–6.4, 0) and (6.4, 0).




45,0 ,
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Example: Graphing a Hyperbola (continued)
2
2
Graph and locate the foci: x  y  1.
36 9
What are the asymptotes?
The foci are  45,0 and 45,0 ,
or (–6.7, 0) and (6.7, 0).
The equations of the asymptotes
1
1
are y  x and y   x.
2
2




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Translations of Hyperbolas – Standard Forms of
Equations of Hyperbolas Centered at (h, k)
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Translations of Hyperbolas – Standard Forms of
Equations of Hyperbolas Centered at (h, k) (continued)
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Example: Graphing a Hyperbola Centered at (h, k)
Graph: 4 x2  24 x  9 y 2  90 y  153  0
We begin by completing the square on x and y.
4 x  24 x  9 y  90 y  153  0
2
2
(4 x  24 x)  (9 y  90 y)  153
2

4 x2  6 x 
2
 
 9 y 2  10 y 
  153
4( x2  6 x  9)  9( y 2  10 y  25)  153  36  225
4( x  3)2  9( y  5)2  36
2
2
2
2
4( x  3) 9( y  5) 36
( x  3) ( y  5)




1
36
36
36
9
4
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Example: Graphing a Hyperbola Centered at (h, k)
(continued)
Graph: 4 x2  24 x  9 y 2  90 y  153  0
( x  3)2 ( y  5) 2
( y  5)2 ( x  3) 2


1 

1
9
4
4
9
We will graph
( y  5)2 ( x  3) 2

1
4
9
The center of the graph is (3, –5).
Step 1 Locate the vertices.
2
a  4, a  2.
The vertices are 2 units above and below the center.
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Example: Graphing a Hyperbola Centered at (h, k)
(continued)
2
2
(
y

5)
(
x

3)
Graph:

1
4
9
Step 1 (cont) Locate the vertices.
The center is (3, –5). The vertices are 2 units above and 2
units below the center.
2 units above (3, –5 + 2) = (3, –3)
2 units below (3, –5 – 2) = (3, –7)
The vertices are (3, –3) and (3, –7).
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Example: Graphing a Hyperbola Centered at (h, k)
(continued)
2
2
(
y

5)
(
x

3)
Graph:

1
4
9
Step 2 Draw a rectangle.
The rectangle passes through points that are 2 units above
and below the center - these points are the vertices, (3, –3)
and (3, –7). The rectangle passes through points that are 3
units to the right and left of the center.
3 units right (3 + 3, –5) = (6, –5)
3 units left (3 – 3, –5) = (0, –5)
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Example: Graphing a Hyperbola Centered at (h, k)
(continued)
2
2
(
y

5)
(
x

3)
Graph:

1
4
9
Step 2 (cont) Draw a rectangle.
The rectangle passes through
(3, –3), (3, –7),
(0, –5), and (6, –5).
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Example: Graphing a Hyperbola Centered at (h, k)
(continued)
2
2
(
y

5)
(
x

3)
Graph:

1
4
9
Step 3 Draw extended diagonals
of the rectangle to obtain the
asymptotes.
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Example: Graphing a Hyperbola Centered at (h, k)
(continued)
2
2
(
y

5)
(
x

3)
Graph:

1
4
9
Step 3 (cont) Draw extended
diagonals of the rectangle
to obtain the asymptotes.
The asymptotes for the unshifted
2
hyperbola are y   x.
3
The asymptotes for the shifted
hyperbola are
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Example: Graphing a Hyperbola Centered at (h, k)
(continued)
2
2
(
y

5)
(
x

3)
Graph:

1
4
9
Step 3 (cont)
2
The asymptotes for the unshifted hyperbola are y   x.
3
The asymptotes for the shifted hyperbola are
2
y  5   ( x  3).
3
2
2
2
y  5  ( x  3)
y5 x2
y  x7
3
3
3
2
y  5   ( x  3)
3
2
y5  x2
3
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y   x3
3
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Example: Graphing a Hyperbola Centered at (h, k)
(continued)
2
2
(
y

5)
(
x

3)
Graph:

1
4
9
Step 4 Draw the two branches
of the hyperbola by starting at
each vertex and approaching
the asymptotes.
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Example: Graphing a Hyperbola Centered at (h, k)
(continued)
2
2
(
y

5)
(
x

3)
Graph:

1
4
9
The foci are at (3, –5 – c) and (3, –5 + c).
c2  a 2  b2
c  4  9  13
2
c   13




The foci are 3, 5  13 and 3, 5  13 .
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Example: Graphing a Hyperbola Centered at (h, k)
(continued)
2
2
(
y

5)
(
x

3)
Graph:

1
4
9
The center is (3, –5).
The vertices are (3, –3)
and (3, –7). The foci are
3, 5  13 and 3, 5  13 .
The asymptotes are
2
y  x7
3
2
and y   x  3.
3




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Example: An Application Involving Hyperbolas
An explosion is recorded by two microphones that are 2
miles apart. Microphone M1 receives the sound 3 seconds
before microphone M2. Assuming sound travels at 1100
feet per second, determine the possible locations of the
explosion relative to the location of the microphones.
We begin by putting the
microphones in a coordinate
system.
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Example: An Application Involving Hyperbolas
(continued)
We know that M1 received the sound 3 seconds before M2.
Because sound travels at 1100 feet per second, the
difference between the distance from P to M1 and the
distance from P to M2 is 3300 feet. The set of all points P
(or locations of the explosion)
satisfying these conditions
fits the definition of a
hyperbola, with microphones
M1 and M2 at the foci.
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Example: An Application Involving Hyperbolas
(continued)
We will use the standard form of the hyperbola’s equation.
P(x, y), the explosion point, lies on this hyperbola.
x2 y 2
 2 1
2
a b
The differences between the distances is 3300 feet. Thus,
2a = 3300 and a = 1650.
x2
y2
 2 1
2
1650 b
x2
y2
 2 1
2,722,500 b
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Example: An Application Involving Hyperbolas
(continued)
The distance from the center, (0, 0), to either focus
(–5280, 0) or (5280, 0) is 5280. Thus, c = 5280.
c2  a 2  b2
52802  16502  b 2
b2  52802  16502  25,155,900
The equation of the hyperbola with a microphone at each
focus is
x2
y2

 1.
2,722,500 25,155,900
We can conclude that the explosion occurred somewhere
on the right branch (the branch closer to M1) of the
hyperbola given by this equation.
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