ohhs pre-calculus mr. j. focht. 8.3 hyperbolas geometry of a hyperbola translations of hyperbolas...
TRANSCRIPT
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OHHS
Pre-Calculus
Mr. J. Focht
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8.3 Hyperbolas
• Geometry of a Hyperbola
• Translations of Hyperbolas
• Eccentricity
8.3
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A Hyperbola is a Conic Section
8.3
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Hyperbola Definition
• Set of all points whose difference of the distances to 2 fixed points is constant.
Focus Focus
(x,y)
d1
d2
d1 – d2 is the same for whatever (x,y) you choose on the blue curves.
8.3
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Hyperbola Terms
Focus FocusCenter
Transverse Axis
Conjugate Axis
Vertex
Vertex
Asymptote
Asymptote
Focal Axis
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Hyperbola Terms
Focus FocusCenter
Asymptote
Asymptote
a = distance from center to vertex
a
b
b = distance from vertex to asymptote
c = distance from center to focus
c
c2 = a2 + b2
8.3
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Hyperbola Equation
Focus FocusCenter
Asymptote
Asymptote
a
bc
(h,k)
1)()(
2
2
2
2
b
ky
a
hx
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Hyperbola Equation
Fo
cus
Fo
cus
Asy
mpt
ote
Asy
mpt
ote
ab
c (h,k
)
1)()(
2
2
2
2
b
hx
a
ky
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Asymptote Equations
Focus FocusCenter
Asymptote
Asymptote
a
bc
(h,k)
)( hxa
bky
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Asymptote Equations
Fo
cus
Fo
cus
Asy
mpt
ote
Asy
mpt
ote
ab
c (h,k
)
)( hxb
aky
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Example
• Find the vertices and the foci of the hyperbola 4x2 - 9y2 = 36.
• Divide by sides by 36.
1)()(
2
2
2
2
b
ky
a
hx
2 24 9 36
36 36 36
x y
2 2
19 4
x y
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Example: Find the Center, Vertices, and Foci
2 2
19 4
x y
a2 = 9
a = 3
Vertices (-3, 0),
(3, 0)
(h, k) = (0, 0)
b2 = 4
c2 = a2 + b2 = 9 + 4 = 13
Foci 0,13
13,0
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Now You Try
1716
22
yx
Find the center, vertices, and foci
8.3
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Find the equation of the hyperbola
4
(1,-5)
(1, 1)
The center is halfway between the foci.
(1, -2)2
2
a
1)()(
2
2
2
2
b
hx
a
ky 1+2
4
c = distance from center to a focus
c = 3
c2 = a2 + b2
9 = 4 + b2
b2 = 5
5
8.3
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Now You Try
• P. 663, #23: Find the equation that satisfies these conditions: Foci (±3, 0), transverse axis length 4
8.3
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Example
• Find the coordinates of the center, foci, and vertices, and the equations of the asymptotes of the graph of
4x2 – y2 + 24x + 4y + 28 = 0
4x2 + 24x – y2 + 4y = -28
4(x2 + 6x ) - (y2 -4y ) = -28
4(x+3)2 – (y-2)2 = 4
+ 9+ 36
+ 4
- 4
8.3
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Example
4(x+3)2 – (y-2)2 = 4
14
2)(y
1
3)(x 22
a = 1 b = 2
c2 = a2 + b2
c2 = 1 + 4
c2 = 5
5c Now let’s find the vertices, foci, and asymptotes on the graph.
8.3
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Example
5c 1
4
2)(y
1
3)(x 22
a=1 b=2(-3, 2)
(-2, 2)(-4, 2)
2),53(
2),53(
h)(xa
bky 3)(x
1
22y
8.3
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Now You Try
• P. 664, #49: Find the center, vertices, and foci of 9x2 – 4y2 – 36x + 8y – 4 =0
8.3
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Eccentricity
• Hyperbolas have eccentricities too.
a
cE Since c > a, E > 1
8.3
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Example
• Write the equation of the hyperbola with center at (-2, -4) , a focus of (2,-4) and eccentricity
3
4
1b
4)(y
a
2)(x2
2
2
2
c = distance from center to focus = 4
a
c
3
4E Since c= 4, a = 3
c2 = a2 + b2 42 = 32 + b2 b2 = 7
9 7
8.3
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Now You Try
• p. 663, #37: Find an equation in standard form for the hyperbola that satisfies the conditions: Center(-3,6), a=5, e=2, vertical transverse axis
8.3
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Home Work
• P. 663-665, #2, 6, 24, 28, 32, 38, 44, 50, #63-68
8.3
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