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( y 6 ) 2 36 ( x + 1 ) 2 16 = 1

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( x 2 ) 2 49 ( y + 7 ) 2 49 = 1

( x 2 ) 2 7 2 ( y + 7 ) 2 7 2 = 1 ; vertices: ( 9 , 7 ) , ( 5 , 7 ) ; foci: ( 2 + 7 2 , 7 ) , ( 2 7 2 , 7 ) ; asymptotes: y = x 9 , y = x 5

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4 x 2 8 x 9 y 2 72 y + 112 = 0

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9 x 2 54 x + 9 y 2 54 y + 81 = 0

( x + 3 ) 2 3 2 ( y 3 ) 2 3 2 = 1 ; vertices: ( 0 , 3 ) , ( 6 , 3 ) ; foci: ( 3 + 3 2 , 1 ) , ( 3 3 2 , 1 ) ; asymptotes: y = x + 6 , y = x

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4 x 2 24 x 36 y 2 360 y + 864 = 0

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4 x 2 + 24 x + 16 y 2 128 y + 156 = 0

( y 4 ) 2 2 2 ( x 3 ) 2 4 2 = 1 ; vertices: ( 3 , 6 ) , ( 3 , 2 ) ; foci: ( 3 , 4 + 2 5 ) , ( 3 , 4 2 5 ) ; asymptotes: y = 1 2 ( x 3 ) + 4 , y = 1 2 ( x 3 ) + 4

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4 x 2 + 40 x + 25 y 2 100 y + 100 = 0

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x 2 + 2 x 100 y 2 1000 y + 2401 = 0

( y + 5 ) 2 7 2 ( x + 1 ) 2 70 2 = 1 ; vertices: ( 1 , 2 ) , ( 1 , 12 ) ; foci: ( 1 , 5 + 7 101 ) , ( 1 , 5 7 101 ) ; asymptotes: y = 1 10 ( x + 1 ) 5 , y = 1 10 ( x + 1 ) 5

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9 x 2 + 72 x + 16 y 2 + 16 y + 4 = 0

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4 x 2 + 24 x 25 y 2 + 200 y 464 = 0

( x + 3 ) 2 5 2 ( y 4 ) 2 2 2 = 1 ; vertices: ( 2 , 4 ) , ( 8 , 4 ) ; foci: ( 3 + 29 , 4 ) , ( 3 29 , 4 ) ; asymptotes: y = 2 5 ( x + 3 ) + 4 , y = 2 5 ( x + 3 ) + 4

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For the following exercises, find the equations of the asymptotes for each hyperbola.

( x 3 ) 2 5 2 ( y + 4 ) 2 2 2 = 1

y = 2 5 ( x 3 ) 4 , y = 2 5 ( x 3 ) 4

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( y 3 ) 2 3 2 ( x + 5 ) 2 6 2 = 1

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9 x 2 18 x 16 y 2 + 32 y 151 = 0

y = 3 4 ( x 1 ) + 1 , y = 3 4 ( x 1 ) + 1

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16 y 2 + 96 y 4 x 2 + 16 x + 112 = 0

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Graphical

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.

( y + 5 ) 2 9 ( x 4 ) 2 25 = 1

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( x 2 ) 2 8 ( y + 3 ) 2 27 = 1

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( y 3 ) 2 9 ( x 3 ) 2 9 = 1

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4 x 2 8 x + 16 y 2 32 y 52 = 0

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x 2 8 x 25 y 2 100 y 109 = 0

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x 2 + 8 x + 4 y 2 40 y + 88 = 0

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64 x 2 + 128 x 9 y 2 72 y 656 = 0

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16 x 2 + 64 x 4 y 2 8 y 4 = 0

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100 x 2 + 1000 x + y 2 10 y 2575 = 0

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4 x 2 + 16 x 4 y 2 + 16 y + 16 = 0

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For the following exercises, given information about the graph of the hyperbola, find its equation.

Vertices at ( 3 , 0 ) and ( −3 , 0 ) and one focus at ( 5 , 0 ) .

x 2 9 y 2 16 = 1

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Vertices at ( 0 , 6 ) and ( 0 , −6 ) and one focus at ( 0 , −8 ) .

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Vertices at ( 1 , 1 ) and ( 11 , 1 ) and one focus at ( 12 , 1 ) .

( x 6 ) 2 25 ( y 1 ) 2 11 = 1

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Center: ( 0 , 0 ) ; vertex: ( 0 , −13 ) ; one focus: ( 0 , 313 ) .

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Center: ( 4 , 2 ) ; vertex: ( 9 , 2 ) ; one focus: ( 4 + 26 , 2 ) .

( x 4 ) 2 25 ( y 2 ) 2 1 = 1

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Center: ( 3 , 5 ) ; vertex: ( 3 , 11 ) ; one focus: ( 3 , 5 + 2 10 ) .

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For the following exercises, given the graph of the hyperbola, find its equation.

y 2 9 ( x + 1 ) 2 9 = 1

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( x + 3 ) 2 25 ( y + 3 ) 2 25 = 1

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Extensions

For the following exercises, express the equation for the hyperbola as two functions, with y as a function of x . Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.

y 2 9 x 2 1 = 1

y ( x ) = 3 x 2 + 1 , y ( x ) = 3 x 2 + 1

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( x 2 ) 2 16 ( y + 3 ) 2 25 = 1

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4 x 2 16 x + y 2 2 y 19 = 0

y ( x ) = 1 + 2 x 2 + 4 x + 5 , y ( x ) = 1 2 x 2 + 4 x + 5

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4 x 2 24 x y 2 4 y + 16 = 0

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Real-world applications

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.

The hedge will follow the asymptotes y = x  and  y = x , and its closest distance to the center fountain is 5 yards.

x 2 25 y 2 25 = 1

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The hedge will follow the asymptotes y = 2 x  and  y = −2 x , and its closest distance to the center fountain is 6 yards.

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The hedge will follow the asymptotes y = 1 2 x and y = 1 2 x , and its closest distance to the center fountain is 10 yards.

x 2 100 y 2 25 = 1

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The hedge will follow the asymptotes y = 2 3 x and y = 2 3 x , and its closest distance to the center fountain is 12 yards.

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The hedge will follow the asymptotes   y = 3 4 x  and  y = 3 4 x , and its closest distance to the center fountain is 20 yards.

x 2 400 y 2 225 = 1

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For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.

The object enters along a path approximated by the line y = x 2 and passes within 1 au (astronomical unit) of the sun at its closest approach, so that the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line y = x + 2.

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The object enters along a path approximated by the line y = 2 x 2 and passes within 0.5 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line y = −2 x + 2.

( x 1 ) 2 0.25 y 2 0.75 = 1

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The object enters along a path approximated by the line y = 0.5 x + 2 and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line y = −0.5 x 2.

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The object enters along a path approximated by the line y = 1 3 x 1 and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line   y = 1 3 x + 1.

( x 3 ) 2 4 y 2 5 = 1

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The object It enters along a path approximated by the line y = 3 x 9 and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line y = −3 x + 9.

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Questions & Answers

How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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