9.5 Notes – Hyperbolas
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9.5 Notes – Hyperbolas
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Hyperbolas: the set of all points for which the difference of the distances to two foci is a constant.
d1
d2 – d1= constant
(x, y)
focus focus
d2
center
The imaginary line between the focal points is the ‘transverse’ axis of the hyperbola.
transverse
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asymptote
focus focus
(c, 0)
ca
vertex
Horizontal transverse axis
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asymptotefocus
(c, 0)
vertex ca
Vertical transverse axis
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A hyperbola can be graphed by locating the vertices (using the a distance from the center) and drawing the two asymptotes through the center of the hyperbola. The foci can be located by using the formula: .
c 2 a2 b2
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Standard Form of equation for a hyperbola(note the a2 is always in the lead term)
Horizontal Transverse axis Vertical Transverse axis
Asymptote: Asymptote:
Foci:
ba b
a
(x h)2
a2 (y k)2
b2 1
(y k)2
a2 (x h)2
b2 1
c 2 a2 b2
by x h k
a a
y x h kb
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Ex 1: State if the hyperbola is horizontal/vertical, find the center, and the eqn of asymptotes.
x 2
9y 2
161
a)
horizontal
Center: (0, 0)4
3y x
by x h k
a
40 0
3y x
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Ex 1: State if the hyperbola is horizontal/vertical, find the center, and the eqn of asymptotes.
b)
vertical
Center: (-2, 1)
(y 1)2
49
(x 2)2
91
ay x h k
b
72 1
3y x
7 17 7 11
3 3 3 3y x y x
7 141
3 3y x
7 141
3 3y x
7 72 1 2 1
3 3y x y x
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Ex 2: Graph each hyperbola by filling in the missing information
2 2
14 25
x y a)
Horizontal or Vertical
center: ( , )
transverse axis(eq):
vertices: ( , ) ( , )
c = ______
foci: ( , ) ( , )
Asymp:
0 0y = 0
2 0 -2 0
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c 2 a2 b22 2
14 25
x y
2 4 25c 2 29c
29 5.4c
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Ex 2: Graph each hyperbola by filling in the missing information
2 2
14 25
x y a)
Horizontal or Vertical
center: ( , )
transverse axis(eq):
vertices: ( , ) ( , )
c = ______
foci: ( , ) ( , )
Asymp:
0 0y = 0
5.45.4 0 -5.4 0
2 0 -2 0
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2 2
14 25
x y b
y x h ka
250 0
4y x
25
4y x
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Ex 2: Graph each hyperbola by filling in the missing information
2 2
14 25
x y a)
Horizontal or Vertical
center: ( , )
transverse axis(eq):
vertices: ( , ) ( , )
c = ______
foci: ( , ) ( , )
Asymp:
0 0y = 0
5.4-5.4 0 5.4 0
25
4y x
2 0 -2 0
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2 22 36 2 36y x
Ex 2: Graph each hyperbola by filling in the missing information
2 22 36 2 36y x
2 22 2
136 1
y x
36 36 36
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Ex 2: Graph each hyperbola by filling in the missing information
Horizontal or Vertical
center: ( , )
transverse axis(eq):
vertices: ( , ) ( , )
c = ______
foci: ( , ) ( , )
Asymp:
-2 2x = -2
-2 8 -2 -4
2 22 2
136 1
y x
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c 2 a2 b2
2 36 1c 2 37c
37 6.1c
2 22 2
136 1
y x
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Ex 2: Graph each hyperbola by filling in the missing information
Horizontal or Vertical
center: ( , )
transverse axis(eq):
vertices: ( , ) ( , )
c = ______
foci: ( , ) ( , )
Asymp:
-2 2x = -2
-2 8 -2 -4
2 22 2
136 1
y x
6.1-2 8.1 -2 -4.1
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ay x h k
b
62 2
1y x
2 22 2
136 1
y x
6 2 2y x 6 2 2y x
6 12 2y x 6 12 2y x
6 14y x 6 10y x
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Ex 2: Graph each hyperbola by filling in the missing information
Horizontal or Vertical
center: ( , )
transverse axis(eq):
vertices: ( , ) ( , )
c = ______
foci: ( , ) ( , )
Asymp:
-2 2x = -2
-2 8 -2 -4
2 22 2
136 1
y x
6.1-2 8.1 -2 -4.1
6 14y x 6 10y x
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Ex 3: Write the equation of the hyperbola centered at the origin with foci (-4, 0) (4, 0) and vertices (-3, 0) and (3, 0)
(x h)2
a2 (y k)2
b2 1
2 2
2
( 0) ( 0)1
9
x y
b
c 2 a2 b2
2 2 24 3 b 216 9 b
27 b
2 2
19 7
x y
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Ex 4: Write the equation of the hyperbola centered at the origin with foci (0, 2) (0, -2) and vertices (0, 1) and (0, -1)
2 2
2 2
( ) ( )1
y k x h
a b
2 2
2
( 0) ( 0)1
1
y x
b
c 2 a2 b2
2 2 22 1 b 24 1 b
23 b
2 2
11 3
y x
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Ex 5: Write the eqn of the hyperbola with center (-2, 1), vertices at (-2, 5) and (-2, -3) and a b-value of 8.
2 2
2 2
( ) ( )1
y k x h
a b
2 2( 1) ( 2)1
16 64
y x
(-2, 1)
(-2, 5)
(-2, -3)
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Ex 6: Write the equation in standard form: 2 29 8 54 56 0x y x y
2 28 9 54 56x x y y
2 28 ___ 9 6 ___ 56 ___ ___x x y y 16 –169 81
2 24 9 3 9x y
2 24 3
19 1
x y
2 23 4
11 9
y x