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Copyright © 2010 Pearson Education, Inc.
HyperbolasHyperbolas
♦ Find equations of hyperbolasFind equations of hyperbolas
♦ Graph hyperbolasGraph hyperbolas
♦ Learn the reflective property of Learn the reflective property of hyperbolashyperbolas
♦ Translate hyperbolasTranslate hyperbolas
9.39.3
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Hyperbola
The set of all co-planar points whose difference of the distances from two fixed points (foci) are constant.
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Hyperbola
Co-vertices endpoints ofconjugate axis
1)()(
2
2
2
2
b
ky
a
hx
Center: (h, k)
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Hyperbola
1)()(
2
2
2
2
b
hx
a
ky
Co-vertices endpoints ofconjugate axis
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Hyperbola
c2 = a2 + b2
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Slide 7.3 - 14Copyright © 2010 Pearson Education, Inc.
Standard Equation for HyperbolasCentered at (h, k)
The hyperbola with center (h, k), and a horizontal transverse axis satisfies the following equation, where c2 = a2 + b2.
Vertices: (h ± a, k)Foci: (h ± c, k)Asymptotes:
x h 2a2
y k 2
b21
y
b
ax h k
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Slide 7.3 - 15Copyright © 2010 Pearson Education, Inc.
Standard Equation for HyperbolasCentered at (h, k)
The hyperbola with center (h, k), and a vertical transverse axis satisfies the following equation, where c2 = a2 + b2.
Vertices: (h, k ± a)Foci: (h, k ± c)Asymptotes:
y k 2a2
x h 2
b21
y
a
bx h k
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Slide 7.3 - 16Copyright © 2010 Pearson Education, Inc.
Example 1
Sketch the graph of
Label vertices, foci, and asymptotes.
Solution
Equation is in standard form with a = 2 and b = 3. It has a horizontal transverse axis with vertices (±2, 0) Endpoints of conjugate axis are (0, ±3). Find c.
x2
4
y 2
91.
c2 a2 b2 4 9 13, or c 13 3.61
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Slide 7.3 - 17Copyright © 2010 Pearson Education, Inc.
Example 1
Solution continued
foci:
asymptotes:
x2
4
y 2
91
13, 0
y b
ax
y 3
2x
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Slide 7.3 - 18Copyright © 2010 Pearson Education, Inc.
Example 2Find the equation of the hyperbola centered at the origin with a vertical transverse axis of length 6 and focus (0, 5). Also find the equations of its asymptotes.
Solution
Since the hyperbola is centered at the origin with a vertical axis, its equation is
y 2
a2
x2
b21
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Slide 7.3 - 19Copyright © 2010 Pearson Education, Inc.
Example 2Solution continuedTransverse axis has length 6 = 2a, so a = 3. One focus is (0, 5) so c = 5. Find b.
y 2
9
x2
161
b2 c2 a2
b c2 a2 52 32 4
y
a
bx, or y
3
4x
Standard equation is
asymptotes are
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Slide 7.3 - 20Copyright © 2010 Pearson Education, Inc.
Example 5
Graph the hyperbola whose equation is
Label the vertices, foci, and asymptotes.
Solution
Vertical transverse axis. Center: (2, –2)
a2 = 9, b2 = 16 c2 = a2 + b2 = 9 + 16 = 25.
a = 3, b = 4, c = 5
(y 2)2
9
(x 2)2
161.
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Slide 7.3 - 21Copyright © 2010 Pearson Education, Inc.
Example 5
Solution continuedCenter: (2, 2) a = 3, b = 4, c = 5
Vertices 3 units above and below center
(2, 1), (2, –5)
Foci 5 units aboveand below center:
(2, 3), (2, –7)
Asymptotes:
y
3
4(x 2) 2
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Slide 7.3 - 22Copyright © 2010 Pearson Education, Inc.
Example 6Write 9x2 – 18x – 4y2 – 16y = 43 in the standard form for a hyperbola centered at (h, k). Identify the center, vertices and foci.
Solution:
9x2 18x 4y 2 16y 43
9 x2 2x __ 4 y 2 4y __ 43
9(x2 2x 1) 4(y 2 4y 4) 43 9 16
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Slide 7.3 - 23Copyright © 2010 Pearson Education, Inc.
Example 6Solution continued
The center is (1, –2). Because a = 2 and the transverse axis is horizontal, the vertices are (1 ± 2, –2). The foci are (1 ± , –2).
9(x2 2x 1) 4(y 2 4y 4) 43 9 16
9(x 1)2 4(y 2)2 36
(x 1)2
4
(y 2)2
91
13
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Graph the following Hyperbola. Find the vertices, foci and asymptotes
149
)5(
16
)1( 22
yx
Center: (-1, 5)a = 4 in x directionb = 7 in y direction
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Graph the following Hyperbola. Find the vertices, foci and asymptotes.
149
)5(
16
)1( 22
yx
Center: (-1, 5)a = 4 b = 7
a2 + b2 = c2
42 + 72 = c2
65 = c2
c65
16 + 49 = c2
)5,651( )5,651(
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Graph the following Hyperbola. Find the vertices, foci and asymptotes
149
)5(
16
)1( 22
yx
Asymptotes
4
7m
75 ( 1)
4y x
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Graph the following Hyperbola. Find the vertices, foci and asymptotes
149
)5(
16
)1( 22
yx
Asymptotes
4
7m
75 ( 1)
4y x
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Graph the following Hyperbola. Find the vertices, foci and asymptotes
149
)5(
16
)1( 22
yx
Asymptotes:
Center: (-1, 5)
Vertices: (-5, 5) (3, 5)
Co-Vertices: (-1, 12) (-1, -2)
)5,651( Foci:
7( 1) 54
y x
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Homework:pg. 656 1-41 odd
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Precalculus Random Conics HWQ
• Find the standard form of the equation of a parabola with vertex
(-2, 1) and directrix at x=1
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Graph the following Hyperbola
136
)4(9
36
)2(4 22
yx
4x2 + 16x - 9y2 + 72y - 5 = 874x2 + 16x - 9y2 + 72y = 87 + 5
4(x2 + 4x + 22) - 9(y2 - 8y + (-4)2) = 92 + 16 - 144
4(x + 2)2 - 9(y - 4)2 = -36
19
)2(
4
)4( 22
xy
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Graph the following Hyperbola
19
)2(
4
)4( 22
xy
Center: (-2, 4)a = 2 in y directionb = 3 in x direction
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Graph the following Hyperbola
Center: (-2, 4)a = 2 b = 3
a2 + b2 = c2
22 + 32 = c2
13 = c2
c13
4 + 9 = c2
)134,2(
)134,2(
19
)2(
4
)4( 22
xy
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Graph the following Hyperbola
)134,2(
19
)2(
4
)4( 22
xy
)134,2(
Asymptotes3
2m
))2((3
24 xy
)2(3
24 xy
3
4
3
24 xy
3
16
3
2 xy
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Graph the following Hyperbola
)134,2(
19
)2(
4
)4( 22
xy
)134,2(
Asymptotes3
2m
))2((3
24
xy
)2(3
24
xy
3
4
3
24
xy
3
8
3
2
xy
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Graph the following Hyperbola
)134,2(
19
)2(
4
)4( 22
xy
)134,2(
Asymptotes
Center: (-2, 4)
Vertices: (-2, 6) (-2, 2)
Co-Vertices: (-5, 4) (1, 4)
Length of Transverse axis: 4
Length of Conjugate axis: 6
)134,2( Foci:
3
16
3
2 xy
3
8
3
2
xy
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Slide 7.3 - 37Copyright © 2010 Pearson Education, Inc.
Trajectory of a Comet
One interpretation of an asymptote relates to trajectories of comets as they approach the sun. Comets travel inparabolic, elliptic, or hyperbolic trajectories. If the speed of a comet is too slow, the gravitational pull of the sun will capture the comet in an elliptical orbit.
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Slide 7.3 - 38Copyright © 2010 Pearson Education, Inc.
Trajectory of a Comet
If the speed of the comet is too fast, the comet will pass by the sun once in a hyperbolic trajectory; farther from the sun, gravity becomes weaker and the comet will eventually return to a straight-line trajectory that is determined by the asymptote of the hyperbola.
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Slide 7.3 - 39Copyright © 2010 Pearson Education, Inc.
Trajectory of a Comet
Finally, if the speed is neither too slow nor too fast, the comet will travel in a parabolic path.
In all three cases, the sun is located at a focus of the conic section.
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Slide 7.3 - 40Copyright © 2010 Pearson Education, Inc.
Reflective Property of Hyperbolas
Hyperbolas have an important reflective property. If a hyperbola is rotated about the x-axis, a hyperboloid is formed.
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Slide 7.3 - 41Copyright © 2010 Pearson Education, Inc.
Reflective Property of Hyperbolas
Any beam of light that is directed toward focus F1 will be reflected by the hyperboloid toward focus F2.
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Homework:pg. Conics Worksheet