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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

find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc 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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