10.410.4

Rotation and Systems of Rotation and Systems of Quadratic EquationsQuadratic Equations

Quick ReviewQuick Review

Assume 0 < /2.

1. Given cot 2 3/ 4, find cos 2 .

2. Given cot 2 1/ 3, find cos 2 .

3. Given cot 2 1, find .

4. Given cot 2 1/ 3, find .

5. Given cot 2 3/ 4, find cos .

Quick Review SolutionsQuick Review SolutionsAssume 0 < /2.

1. Given cot 2 3/ 4, find cos 2 .

2. Given cot 2 1/ 3, find cos 2 .

3. Given cot 2 1, find .

4. Given cot 2 1/ 3, find .

5. Given cot 2 3/ 4, find cos .

3/5

1/2

/8

/6

2/ 5

What you’ll learn aboutWhat you’ll learn about Second-Degree Equations in Two VariablesSecond-Degree Equations in Two Variables Translating Axes versus Translating GraphsTranslating Axes versus Translating Graphs Rotation of AxesRotation of Axes Discriminant Test Discriminant Test

… … and whyand whyYou will see ellipses, hyperbolas, and parabolas You will see ellipses, hyperbolas, and parabolas

as as members of the family of conic sections rather members of the family of conic sections rather

than as than as separate types of curves.separate types of curves.

Translation-of-Axes FormulasTranslation-of-Axes Formulas

The coordinates ( , ) and ( ', ') based on parallel sets of axes are

related by either of the following :

' and ' or ' and ' .

x y x y

x x h y y k x x h y y k translations formulas

Example Example Translation FormulaTranslation Formula

2 2Prove that 9 4 18 16 11 0 is the equation of an ellipse.

Translate the coordinate axes so that the origin is at the center of this ellipse.

x y x y

Example Example Translation FormulaTranslation Formula

2 2Prove that 9 4 18 16 11 0 is the equation of an ellipse.

Translate the coordinate axes so that the origin is at the center of this ellipse.

x y x y

2 2

2 2

2 2

2 2

Complete the square for both the and .

9 18 4 16 11

9( 2 1) 4( 4 4) 11 9 16

9( 1) 4( 2) 36

( 1) ( 2)1

4 9This is a standard equation of an ellipse. If we let ' 1

and ' 2, then

x y

x x y y

x x y y

x y

x y

x x

y y

22

the equation of the ellipse becomes

' ( ')1.

4 9

x y

Rotation-of-Axes FormulasRotation-of-Axes Formulas

The coordinates ( , ) and ( ', ') based on rotated sets of axes are

related by either of the following :

' cos sin and ' sin cos , or

'cos 'sin and 'sin 'cos ,

where

x y x y

x x y y x y

x x y y x y

rotation formulas

, 0 / 2, is the . angle of rotation

Rotation of Cartesian Rotation of Cartesian Coordinate AxesCoordinate Axes

Example Example Rotation of AxesRotation of Axes

Prove that 2 25 0 is the equation of a hyperbola by rotating the

coordinate axes through an angle / 4.

xy

Example Example Rotation of AxesRotation of Axes

Prove that 2 25 0 is the equation of a hyperbola by rotating the

coordinate axes through an angle / 4.

xy

2 2

2 2

The rotation equations are

'cos / 4 'sin / 4 and 'sin / 4 'cos / 4

' ' ' ' and .

2 2The equation 2 25 0 becomes

' ' ' '2 25 0

2 2

' ' 25

' '1

25 25

x x y y x y

x y x yx y

xy

x y x y

x y

x y

Coefficients for a Conic in a Coefficients for a Conic in a Rotated SystemRotated System

2 2

If we apply the rotation formulas to the general second-degree equation in

and , we obtain a second-degree equation in ' and ' of the form

' ' ' ' ' ' ' ' ' ' ' ' 0, where the coefficients

x y x y

A x B x y C y D x E y F 2 2

2 2

are

' cos cos sin sin

' cos 2 ( )sin 2

' cos cos sin sin

' cos sin

' cos sin

'

A A B C

B B C A

C C B A

D D E

E E D

F F

Angle of Rotation to Eliminate Angle of Rotation to Eliminate the Cross-Product Termthe Cross-Product Term

If 0, an angle of rotation such that cot 2 and 0 / 2

will eliminate the term ' ' ' from the second degree equation in the

rotated ' ' coordinate system.

A CB

BB x y

x y

Discriminant TestDiscriminant Test

2 2

2

2

2

The second-degree equation 0 graphs as

a hyperbola if 4 0,

a parabola if 4 0,

an ellipse if 4 0,

except for degenerate cases.

Ax Bxy Cy Dx Ey F

B AC

B AC

B AC

Conics and the Equation Conics and the Equation AxAx22++BxyBxy++CyCy22++DxDx++EyEy++FF=0=0