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6-3 Hyperbolas
Unit 6 Conics
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Concepts and Objectives
Hyperbolas (Obj. #21)
Identify equations of hyperbolas From the equation, identify the
center, direction of
opening, vertices,x-radius,y-radius, slope of the
,
Sketch the hyperbola
Determine the eccentricity
Write the equation of the hyperbola
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Conics
A quadratic relation is an equation of the form
The coefficients of the variables determine what shape
+ + + + + =2 2 0 Ax Bxy Cy Dx Ey F
t e grap o t e equat on ta es. Circle: B = 0,A = C, andA, C>
0
Ellipse: B = 0,A C, andA, C> 0
Hyperbola: B = 0,A and Care different signs
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Hyperbolas
Hyperbolas have two disconnected branches. Each
branch approaches diagonal asymptotes. Parts of a hyperbola:
Center
ert ces Asymptotes
Hyperbola
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Hyperbolas
The general equation of a hyperbola is
or
=
22
1
x y
x h y k
r r
+ =
22
1
x y
x h y k
r r
The hyperbola opens in whichever direction has thepositive term
(x-direction ifxis positive,y-direction ify
is positive).
The slope of the asymptotes is always .
The vertices are rxor ryfrom the center, whichever is
positive. a is the positive term radius, b is the negative
term radius.
y
x
r
r
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Hyperbolas
Example: Graph + + + =2 29 4 90 32 197 0 x y x y
( ) ( ) ( ) ( )++ =+ + 22 2 22 29 10 4 8 1975 4 9 5 4 4 x x y
y
( ) ( )+ =
2 2
9 5 4 4 36x y
( ) ( )+ + =
2 2
5 41
4 9
x y
( ) ( )+
+ =
2 2
2 2
5 41
2 3x y
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Hyperbolas
Example: Graph
Center (5, 4)
+ + + =2 29 4 90 32 197 0 x y x y
( ) ( )+ + =2 2
2 2
5 41
2 3
x y
opens iny-directionrx= 2, ry= 3
vertices 3
slope of asymptotes: 3
2
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Focal Length
In an ellipse, the sum of the distances from a point on the
ellipse to the two foci is constant, but in a hyperbola, itsthe
difference between the distances that is constant.
To find the focal radius, we can use the Pythagorean
.
Notice thatc > a for the
hyperbola. a
bc
= +2 2 2c a b
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Eccentricity
Like the ellipse, the eccentricityof the hyperbola
determines the basic shape, and like the ellipse,
theeccentricity of the hyperbola is
=c
e
In an ellipse, e will always be between 0 and 1, but in a
hyperbola, e will always be greater than 1.
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Eccentricity
Example: Find the eccentricity of the hyperbola
=2 2
19 4
x y
2 2y
a = 3, b = 2
=2 23 2
= +2 2 23 2c= + =9 4 13
= 13c
= 13
1.23
e
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Eccentricity
Example: Write the equation of the hyperbola with
eccentricity 2 and foci at(9, 5) and (3, 5).
The focis coordinates tell us that the hyperbola opens in
- , , , .
=3
2a
= 1.5a
= =2 9 2.25 6.75b
=2 2.25a
( ) ( )+ =
2 2
6 51
2.25 6.75
x y ( ) ( )+ =
2 2
4 6 4 51
9 27
x y
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Homework
Algebra & Trigonometry(green book)
Page 486: 3-15 (3s) Turn in: 6, 12
College Algebra (brown book) Page 978: 27-48 (3s)
Turn in: 30, 36, 42, 45
