<< 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

if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
Rahul Reply
sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask