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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

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
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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