<< Chapter < Page Chapter >> Page >

( y 1 ) 2 49 ( x + 1 ) 2 4 = 1


Got questions? Get instant answers now!

x 2 4 y 2 + 6 x + 32 y 91 = 0

Got questions? Get instant answers now!

2 y 2 x 2 12 y 6 = 0


Got questions? Get instant answers now!

For the following exercises, find the equation of the hyperbola.

Center at ( 0 , 0 ) , vertex at ( 0 , 4 ) , focus at ( 0 , −6 )

Got questions? Get instant answers now!

Foci at ( 3 , 7 ) and ( 7 , 7 ) , vertex at ( 6 , 7 )

( x 5 ) 2 1 ( y 7 ) 2 3 = 1

Got questions? Get instant answers now!

The Parabola

For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.

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

( x + 2 ) 2 = 1 2 ( y 1 ) ; vertex: ( −2 , 1 ) ; focus: ( −2 , 9 8 ) ; directrix: y = 7 8

Got questions? Get instant answers now!

y 2 6 y 6 x 3 = 0

Got questions? Get instant answers now!

x 2 + 10 x y + 23 = 0

( x + 5 ) 2 = ( y + 2 ) ; vertex: ( 5 , 2 ) ; focus: ( 5 , 7 4 ) ; directrix: y = 9 4

Got questions? Get instant answers now!

For the following exercises, graph the parabola, labeling vertex, focus, and directrix.

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


Got questions? Get instant answers now!

x 2 8 x 10 y + 46 = 0

Got questions? Get instant answers now!

2 y 2 + 12 y + 6 x + 15 = 0


Got questions? Get instant answers now!

For the following exercises, write the equation of the parabola using the given information.

Focus at ( −4 , 0 ) ; directrix is x = 4

Got questions? Get instant answers now!

Focus at ( 2 , 9 8 ) ; directrix is y = 7 8

( x 2 ) 2 = ( 1 2 ) ( y 1 )

Got questions? Get instant answers now!

A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 5 feet across at its opening and 1.5 feet deep.

Got questions? Get instant answers now!

Rotation of Axes

For the following exercises, determine which of the conic sections is represented.

16 x 2 + 24 x y + 9 y 2 + 24 x 60 y 60 = 0

B 2 4 A C = 0 , parabola

Got questions? Get instant answers now!

4 x 2 + 14 x y + 5 y 2 + 18 x 6 y + 30 = 0

Got questions? Get instant answers now!

4 x 2 + x y + 2 y 2 + 8 x 26 y + 9 = 0

B 2 4 A C = 31 < 0 , ellipse

Got questions? Get instant answers now!

For the following exercises, determine the angle θ that will eliminate the x y term, and write the corresponding equation without the x y term.

x 2 + 4 x y 2 y 2 6 = 0

Got questions? Get instant answers now!

x 2 x y + y 2 6 = 0

θ = 45 , x 2 + 3 y 2 12 = 0

Got questions? Get instant answers now!

For the following exercises, graph the equation relative to the x y system in which the equation has no x y term.

9 x 2 24 x y + 16 y 2 80 x 60 y + 100 = 0

Got questions? Get instant answers now!

x 2 x y + y 2 2 = 0

θ = 45

Got questions? Get instant answers now!

6 x 2 + 24 x y y 2 12 x + 26 y + 11 = 0

Got questions? Get instant answers now!

Conic Sections in Polar Coordinates

For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.

r = 10 1 5   cos   θ

Hyperbola with e = 5 and directrix 2 units to the left of the pole.

Got questions? Get instant answers now!

r = 6 3 + 2   cos   θ

Got questions? Get instant answers now!

r = 1 4 + 3   sin   θ

Ellipse with e = 3 4 and directrix 1 3 unit above the pole.

Got questions? Get instant answers now!

r = 3 5 5   sin   θ

Got questions? Get instant answers now!

For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.

r = 8 4 + 3   sin   θ

Got questions? Get instant answers now!

r = 10 4 + 5   cos   θ


Got questions? Get instant answers now!

r = 9 3 6   cos   θ

Got questions? Get instant answers now!

For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.

Directrix is x = 3 and eccentricity e = 1

r = 3 1 + cos     θ

Got questions? Get instant answers now!

Directrix is y = −2 and eccentricity e = 4

Got questions? Get instant answers now!

Practice test

For the following exercises, write the equation in standard form and state the center, vertices, and foci.

x 2 9 + y 2 4 = 1

x 2 3 2 + y 2 2 2 = 1 ; center: ( 0 , 0 ) ; vertices: ( 3 , 0 ) , ( –3 , 0 ) , ( 0 , 2 ) , ( 0 , −2 ) ; foci: ( 5 , 0 ) , ( 5 , 0 )

Got questions? Get instant answers now!

9 y 2 + 16 x 2 36 y + 32 x 92 = 0

Got questions? Get instant answers now!

For the following exercises, sketch the graph, identifying the center, vertices, and foci.

( x 3 ) 2 64 + ( y 2 ) 2 36 = 1

center: ( 3 , 2 ) ; vertices: ( 11 , 2 ) , ( −5 , 2 ) , ( 3 , 8 ) , ( 3 , −4 ) ; foci: ( 3 + 2 7 , 2 ) , ( 3 2 7 , 2 )

Got questions? Get instant answers now!

2 x 2 + y 2 + 8 x 6 y 7 = 0

Got questions? Get instant answers now!

Write the standard form equation of an ellipse with a center at ( 1 , 2 ) , vertex at ( 7 , 2 ) , and focus at ( 4 , 2 ).

( x 1 ) 2 36 + ( y 2 ) 2 27 = 1

Got questions? Get instant answers now!

A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet away from the wall, how high should the ceiling be?

Got questions? Get instant answers now!

For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.

x 2 49 y 2 81 = 1

x 2 7 2 y 2 9 2 = 1 ; center: ( 0 , 0 ) ; vertices ( 7 , 0 ) , ( −7 , 0 ) ; foci: ( 130 , 0 ) , ( 130 , 0 ) ; asymptotes: y = ± 9 7 x

Got questions? Get instant answers now!

16 y 2 9 x 2 + 128 y + 112 = 0

Got questions? Get instant answers now!

For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.

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

center: ( 3 , −3 ) ; vertices: ( 8 , −3 ) , ( −2 , −3 ) ; foci: ( 3 + 26 , −3 ) , ( 3 26 , −3 ) ; asymptotes: y = ± 1 5 ( x 3 ) 3

Got questions? Get instant answers now!

y 2 x 2 + 4 y 4 x 18 = 0

Got questions? Get instant answers now!

Write the standard form equation of a hyperbola with foci at ( 1 , 0 ) and ( 1 , 6 ) , and a vertex at ( 1 , 2 ) .

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

Got questions? Get instant answers now!

For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.

3 x 2 12 x y + 11 = 0

( x 2 ) 2 = 1 3 ( y + 1 ) ; vertex: ( 2 , −1 ) ; focus: ( 2 , 11 12 ) ; directrix: y = 13 12

Got questions? Get instant answers now!

For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.

( x 1 ) 2 = −4 ( y + 3 )

Got questions? Get instant answers now!

y 2 + 8 x 8 y + 40 = 0


Got questions? Get instant answers now!

Write the equation of a parabola with a focus at ( 2 , 3 ) and directrix y = −1.

Got questions? Get instant answers now!

A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?

Approximately 8.49 feet

Got questions? Get instant answers now!

For the following exercises, determine which conic section is represented by the given equation, and then determine the angle θ that will eliminate the x y term.

3 x 2 2 x y + 3 y 2 = 4

Got questions? Get instant answers now!

x 2 + 4 x y + 4 y 2 + 6 x 8 y = 0

parabola; θ 63.4

Got questions? Get instant answers now!

For the following exercises, rewrite in the x y system without the x y term, and graph the rotated graph.

11 x 2 + 10 3 x y + y 2 = 4

Got questions? Get instant answers now!

16 x 2 + 24 x y + 9 y 2 125 x = 0

x 2 4 x + 3 y = 0

Got questions? Get instant answers now!

For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.

r = 5 4 + 6   cos   θ

Hyperbola with e = 3 2 , and directrix 5 6 units to the right of the pole.

Got questions? Get instant answers now!

For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci.

r = 12 4 8   sin   θ

Got questions? Get instant answers now!

r = 2 4 + 4   sin   θ

Got questions? Get instant answers now!

Find a polar equation of the conic with focus at the origin, eccentricity of e = 2 , and directrix: x = 3.

Got questions? Get instant answers now!

Questions & Answers

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
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
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
Practice Key Terms 2

Get the best College algebra course in your pocket!





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