chapter 9 conic sections and analytic geometry copyright © 2014, 2010, 2007 pearson education, inc....

Chapter 9Conic Sections andAnalytic Geometry

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

9.4 Rotation of Axes


Objectives:

•Identify conics without completing the square.•Use rotation of axes formulas.•Write equations of rotated conics in standard form.•Identify conics without rotating axes.


Identifying Conic Sections without Completing the Square

Conic sections can be represented both geometrically (as intersecting planes and cones) and algebraically. The equations of the conic sections we have considered in the first three sections of this chapter can be expressed in the form

in which A and C are not both zero. We can identify a conic section without completing the square by comparing the values of A and C.

2 2 0,Ax Cy Dx Ey F


Identifying a Conic Section without Completing the Square (continued)


Example: Identifying a Conic Section without Completing the Square

Identify the graph of the following nondegenerate conic section:

2 23 2 12 4 2 0x y x y 3, 2A C

3(2) 6AC

Because A is not equal to C and AC is positive, the graph of the equation is an ellipse.




2 2 6 3 0x y x y 1, 1A C

Because A and C are equal, the graph of the equation is a circle.




2 12 4 52 0y x y 0, 1A C

0(1) 0AC

Because AC = 0, the graph of the equation is a parabola.




2 29 16 90 64 17 0x y x y 9, 16A C

9( 16) 144AC

Because AC is negative, the graph of the equation is an hyperbola.


Rotation of Axes

Except for degenerate cases, the general second-degree equation

represents one of the conic sections. Due to the xy-term in the equation, these conic sections are rotated in such a way that their axes are no longer parallel to the x- and y-axes. To reduce these equations to forms of the conic sections with which we are already familiar, we use a procedure called rotation of axes.

2 2 0Ax Bxy Cy Dx Ey F


Rotation of Axes Formula


Example: Rotating Axes

Write the equation xy = 2 in terms of a rotated x′y′-system if the angle of rotation from the x-axis to the x′-axis is 45°. Express the equation in standard form. Use the rotated system to graph xy = 2.

cos sin x x y cos45 sin 45 x y

2 22 2

x y 22

x y


Example: Rotating Axes (continued)


sin cos y x y

2 22 2

x y 22

x y

sin 45 cos45x y




2xy

2 22

2 2

x y x y

22

4 x y x y

2 212

2 x y

2 2

22 2

x y

2 2

14 4

x y




The graph of xy = 2 or2 2

14 4

x y

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

yx

y

vertex(2, 0)

vertex(–2, 0)


Using Rotations to Transform Equations with xy-Terms to Standard Equations of Conic Sections

A rotation of axes through an appropriate angle can transform the equation to one of the standard forms of the conic sections in x′ and y′ in which no x′y′-term appears.


Amount of Rotation Formula


Writing the Equation of a Rotated Conic in Standard Form


Example: Writing the Equation of a Rotated Conic Section in Standard Form

Rewrite the equation

in a rotated x′y′-system without an x′y′-term. Express the equation in the standard form of a conic section. Graph the conic section in the rotated system.

2 22 3 2 0x xy y

Step 1 Use the given equation to find cot 2 .

2 22 3 2 0x xy y 2, 3, 1A B C

cot 2A C

B 2 1

3

1 333


Example: Writing the Equation of a Rotated Conic Section in Standard Form (continued)



2 22 3 2 0x xy y

Step 2 Use the expression for to determinethe angle of rotation.

cot 2

3cot 2

3 2 60 30





2 22 3 2 0x xy y

Step 3 Substitute in the rotation formulas and simplify.

cos sin x x y cos30 sin30 x y

3 12 2

x y

32

x y





2 22 3 2 0x xy y

Step 3 (cont) Substitute in the rotation formulas and simplify.

sin cos y x y sin30 cos30 x y

1 32 2

x y

32

x y



Step 4 Substitute the expressions for x and y from the rotation formulas in the given equation and simplify.

32

x yx

32

x yy

2 22 3 2 0x xy y 2

2

3 3 32 3

2 2 2

32 0

2

x y x y x y

x y



Step 4 (cont) Substitute the expressions for x and y from the rotation formulas in the given equation and simplify.

2 2 2 2

2 2

3 2 3 3 3 32 3

4 4

2 3 32 0

4

x x y y x x y x y y

x x y y

2 2 2 2

2 2

2 3 2 3 3 3 2 3

2 3 3 8 0

x x y y x x y y

x x y y




2 2 2

2 2 2

6 4 3 2 3 3 3 3

3 2 3 3 8 0

x x y y x x y x y

y x x y y

2 2 2

2 2 2

6 3 4 3 3 3 3

2 3 2 3 3 8 0

x x x x y x y x y

x y y y y

2 210 2 8 0 x y




2 210 2 8 x y

2 2

14 45

x y

Step 5 Write the equation involving x′ and y′ in standard form.

2 210 2 8 x y



Rewrite the equation in a rotated x′y′-system without an x′y′-term. Express the equation in the standard form of a conic section. Graph the conic section in the rotated system.

2 22 3 2 0x xy y

2 2

14 45

x y

2 22 3 2 0x xy y

Graph of

or

xy

(0,2)

(0, 2)

(0.9,0)

( 0.9,0)

Major axis y′


Identifying Conic Sections without Rotating Axes


Example: Identifying a Conic Section without Rotating Axes

Identify the graph of 2 23 2 3 2 2 3 0.x xy y x y

2 23 2 3 2 2 3 0x xy y x y 3, 2 3, 1A B C

22 4 2 3 4(3)(1)B AC 4 3 12 12 12 0

Because the equation is a parabola.2 4 0,B AC

chapter 9 conic sections and analytic geometry copyright © 2014, 2010, 2007 pearson education, inc....

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