<< Chapter < Page Chapter >> Page >

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
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

x 2 10 x + 4 y 10 = 0

A B = 0 , parabola

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

4 x 2 y 2 + 8 x 1 = 0

A B = 4 < 0 , hyperbola

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

A B = 6 > 0 , ellipse

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

B 2 4 A C = 0 , parabola

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

B 2 4 A C = 0 , parabola

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

B 2 4 A C = 96 < 0 , ellipse

Got questions? Get instant answers now!

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°

Got questions? Get instant answers now!

4 x 2 x y + 4 y 2 2 = 0 , θ = 45°

7 x 2 + 9 y 2 4 = 0

Got questions? Get instant answers now!

2 x 2 + 8 x y 1 = 0 , θ = 30°

Got questions? Get instant answers now!

2 x 2 + 8 x y + 1 = 0 , θ = 45°

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

Got questions? Get instant answers now!

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

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

θ = 60 , 11 x 2 y 2 + 3 x + y 4 = 0

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

−3 x 2 3 x y 2 y 2 x = 0

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

θ = 45 , 3 x 2 y 2 2 x + 2 y + 1 = 0

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

x 2 4 + y 2 1 = 1 , θ = 45

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

Got questions? Get instant answers now!

y 2 16 + x 2 9 = 1 , θ = 45

Got questions? Get instant answers now!

y 2 x 2 = 1 , θ = 45

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

Got questions? Get instant answers now!

x = ( y 1 ) 2 , θ = 30

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

Got questions? Get instant answers now!

x 2 9 + y 2 4 = 1 , θ = 30

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.

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

Got questions? Get instant answers now!

x 2 10 x y + y 2 24 = 0

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

θ = 45

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

θ = 60

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

θ 36.9

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Questions & Answers

sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb
Practice Key Terms 3

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