<< 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

the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
x^20+x^15+x^10+x^5/x^2+1
Urmila Reply
evaluate each algebraic expression. 2x+×_2 if ×=5
Sarch Reply
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
Awe Reply
By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
jaks
Ryan
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
Shahid Reply
Practice Key Terms 3

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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