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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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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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