<< Chapter < Page Chapter >> Page >

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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Graphical

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Questions & Answers

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
what is the meaning
Dominic
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask