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Identify the conic for each of the following without rotating axes.

  1. x 2 9 x y + 3 y 2 12 = 0
  2. 10 x 2 9 x y + 4 y 2 4 = 0
  1. hyperbola
  2. ellipse
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Access this online resource for additional instruction and practice with conic sections and rotation of axes.

Key equations

General Form equation of a conic section A x 2 + B x y + C y 2 + D x + E y + F = 0
Rotation of a conic section x = x cos   θ y sin   θ y = x sin   θ + y cos   θ
Angle of rotation θ , where  cot ( 2 θ ) = A C B

Key concepts

  • Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.
  • A nondegenerate conic section has the general form A x 2 + B x y + C y 2 + D x + E y + F = 0 where A , B and C are not all zero. The values of A , B , and C determine the type of conic. See [link] .
  • Equations of conic sections with an x y term have been rotated about the origin. See [link] .
  • The general form can be transformed into an equation in the x and y coordinate system without the x y term. See [link] and [link] .
  • An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See [link] .

Section exercises

Verbal

What effect does the x y term have on the graph of a conic section?

The x y term causes a rotation of the graph to occur.

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If the equation of a conic section is written in the form A x 2 + B y 2 + C x + D y + E = 0 and A B = 0 , what can we conclude?

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If the equation of a conic section is written in the form A x 2 + B x y + C y 2 + D x + E y + F = 0 , and B 2 4 A C > 0 , what can we conclude?

The conic section is a hyperbola.

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Given the equation a x 2 + 4 x + 3 y 2 12 = 0 , what can we conclude if a > 0 ?

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For the equation A x 2 + B x y + C y 2 + D x + E y + F = 0 , the value of θ that satisfies cot ( 2 θ ) = A C B gives us what information?

It gives the angle of rotation of the axes in order to eliminate the x y term.

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Algebraic

For the following exercises, determine which conic section is represented based on the given equation.

9 x 2 + 4 y 2 + 72 x + 36 y 500 = 0

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x 2 10 x + 4 y 10 = 0

A B = 0 , parabola

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

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4 x 2 y 2 + 8 x 1 = 0

A B = 4 < 0 , hyperbola

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4 y 2 5 x + 9 y + 1 = 0

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2 x 2 + 3 y 2 8 x 12 y + 2 = 0

A B = 6 > 0 , ellipse

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4 x 2 + 9 x y + 4 y 2 36 y 125 = 0

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3 x 2 + 6 x y + 3 y 2 36 y 125 = 0

B 2 4 A C = 0 , parabola

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3 x 2 + 3 3 x y 4 y 2 + 9 = 0

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2 x 2 + 4 3 x y + 6 y 2 6 x 3 = 0

B 2 4 A C = 0 , parabola

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x 2 + 4 2 x y + 2 y 2 2 y + 1 = 0

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8 x 2 + 4 2 x y + 4 y 2 10 x + 1 = 0

B 2 4 A C = 96 < 0 , ellipse

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For the following exercises, find a new representation of the given equation after rotating through the given angle.

3 x 2 + x y + 3 y 2 5 = 0 , θ = 45°

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4 x 2 x y + 4 y 2 2 = 0 , θ = 45°

7 x 2 + 9 y 2 4 = 0

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2 x 2 + 8 x y 1 = 0 , θ = 30°

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2 x 2 + 8 x y + 1 = 0 , θ = 45°

3 x 2 + 2 x y 5 y 2 + 1 = 0

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4 x 2 + 2 x y + 4 y 2 + y + 2 = 0 , θ = 45°

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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 + 3 3 x y + 4 y 2 + y 2 = 0

θ = 60 , 11 x 2 y 2 + 3 x + y 4 = 0

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4 x 2 + 2 3 x y + 6 y 2 + y 2 = 0

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9 x 2 3 3 x y + 6 y 2 + 4 y 3 = 0

θ = 150 , 21 x 2 + 9 y 2 + 4 x 4 3 y 6 = 0

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−3 x 2 3 x y 2 y 2 x = 0

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16 x 2 + 24 x y + 9 y 2 + 6 x 6 y + 2 = 0

θ 36.9 , 125 x 2 + 6 x 42 y + 10 = 0

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x 2 + 4 x y + 4 y 2 + 3 x 2 = 0

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x 2 + 4 x y + y 2 2 x + 1 = 0

θ = 45 , 3 x 2 y 2 2 x + 2 y + 1 = 0

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4 x 2 2 3 x y + 6 y 2 1 = 0

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Graphical

For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.

y = x 2 , θ = 45

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

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x 2 4 + y 2 1 = 1 , θ = 45

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

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y 2 16 + x 2 9 = 1 , θ = 45

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y 2 x 2 = 1 , θ = 45

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

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x = ( y 1 ) 2 , θ = 30

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

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x 2 9 + y 2 4 = 1 , θ = 30

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

x 2 + 10 x y + y 2 6 = 0

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x 2 10 x y + y 2 24 = 0

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4 x 2 3 3 x y + y 2 22 = 0

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6 x 2 + 2 3 x y + 4 y 2 21 = 0

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11 x 2 + 10 3 x y + y 2 64 = 0

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21 x 2 + 2 3 x y + 19 y 2 18 = 0

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16 x 2 + 24 x y + 9 y 2 130 x + 90 y = 0

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

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13 x 2 6 3 x y + 7 y 2 16 = 0

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4 x 2 4 x y + y 2 8 5 x 16 5 y = 0

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For the following exercises, determine the angle of rotation in order to eliminate the x y term. Then graph the new set of axes.

6 x 2 5 3 x y + y 2 + 10 x 12 y = 0

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6 x 2 5 x y + 6 y 2 + 20 x y = 0

θ = 45

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6 x 2 8 3 x y + 14 y 2 + 10 x 3 y = 0

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4 x 2 + 6 3 x y + 10 y 2 + 20 x 40 y = 0

θ = 60

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8 x 2 + 3 x y + 4 y 2 + 2 x 4 = 0

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16 x 2 + 24 x y + 9 y 2 + 20 x 44 y = 0

θ 36.9

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For the following exercises, determine the value of k based on the given equation.

Given 4 x 2 + k x y + 16 y 2 + 8 x + 24 y 48 = 0 , find k for the graph to be a parabola.

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Given 2 x 2 + k x y + 12 y 2 + 10 x 16 y + 28 = 0 , find k for the graph to be an ellipse.

4 6 < k < 4 6

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Given 3 x 2 + k x y + 4 y 2 6 x + 20 y + 128 = 0 , find k for the graph to be a hyperbola.

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Given k x 2 + 8 x y + 8 y 2 12 x + 16 y + 18 = 0 , find k for the graph to be a parabola.

k = 2

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Given 6 x 2 + 12 x y + k y 2 + 16 x + 10 y + 4 = 0 , find k for the graph to be an ellipse.

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Questions & Answers

how can are find the domain and range of a relations
austin Reply
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
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
how to solve the sums
meena
hello guys
meena
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
Practice Key Terms 3

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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