10.5 Rotation of Conics. The Standard Equation for all Conics Ax 2 + Bxy + Cy 2 + Dx + Ey + F = o So far B has equal zero and all graphs have been horizontal.

10.5 Rotation of Conics

The Standard Equation for all Conics

Ax2 + Bxy + Cy2 + Dx + Ey + F = o

So far B has equal zero and all graphs have been horizontal or vertical.

Not anymore.

The Standard Equation for all Conics

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

So far B has equal zero and all graphs have been horizontal or vertical.

Not anymore.

How to Find θRotation off the x axis

Given the Conic BCA

Cot2

2

1

2

342

0235324 22

Cot

yxyxyx

How to Find θRotation through

Given the Conic BCA

Cot2

2

1

2

342

0235324 22

Cot

yxyxyx

uTan

uTan

uCot

uLet

2

221

2

1

How to Find θRotation through

Given the Conic BCA

Cot2

2

1

2

342

0235324 22

Cot

yxyxyx

uTan

uTan

uCot

uLet

2

221

2

1

7.31

4.632

4.63

u

When the graph of the conics is moved x and y are called x‘ and y'

Ellipse rotated 45°

yx,

yx ,

Write x and y in the new graph

Let θ = 30°

cossin

sincos

yxy

yxx

yxyxy

yxyxx

23

21

30cos30sin

21

23

30sin30cos

Find the coordinates of a point in a rotated coordinate system

Given θ = 60° ; (3, 1) Find

yx ,

yxCosySinxy

yxSinyCosxx

2

1

2

36060

2

3

2

16060

yx

yx

2

1

2

31

2

3

2

13

Solve for

System of Equations yx

yx

2

1

2

31

2

3

2

13 yx ,

yx

yx

yx

36

32

36

Solve for

System of Equations yx

yx

2

1

2

31

2

3

2

13 yx ,

yx

yx

yx

36

32

36

2

331

4

362

4362

3362

3632

y

y

yy

yy

Solve for

System of Equations yx

yx

2

1

2

31

2

3

2

13 yx ,

yx

yx

yx

36

32

36

2

33

2

93

2

12

2

936

2

33136

x

x

x

2

331

4

362

4362

3362

3632

y

y

yy

yy

Solve for

System of Equations yx

yx

2

1

2

31

2

3

2

13 yx ,

yx

yx

yx

36

32

36

2

33

2

93

2

12

2

936

2

33136

x

x

x

2

331

4

362

4362

3362

3632

y

y

yy

yy

2

331,

2

33

Classification of Conics by the Discriminant

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

04

04

04

2

2

2

ACBHyperbola

ACBParabola

ACBCircleorEllipse

Classification of Conics by the Discriminant

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

04

04

04

2

2

2

ACBHyperbola

ACBParabola

ACBCircleorEllipse

0457612 22 yxyx

Classification of Conics by the Discriminant

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

04

04

04

2

2

2

ACBHyperbola

ACBParabola

ACBCircleorEllipse

0457612 22 yxyx 3003363671246 2

Ellipse

Finding the intersection of two conics

Given 03

042

22

yx

yx

Finding the intersection of two conics

Given 03

042

22

yx

yx

14

014

04)3(

3

2

2

xorx

xx

xx

yx

Finding the intersection of two conics

Given 03

042

22

yx

yx

1

014

04)3(

3

2

2

x

xx

xx

yx

12

432

2

y

y

Finding the intersection of two conics

Given 03

042

22

yx

yx

1

014

04)3(

3

2

2

x

xx

xx

yx

12

432

2

y

y

3

13 2

y

y

Finding the intersection of two conics

Given 03

042

22

yx

yx

1

014

04)3(

3

2

2

x

xx

xx

yx

12

432

2

y

y

3

13 2

y

y

3,1,3,1

Homework

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