<< 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

what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
Practice Key Terms 2

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask