10.5 Rotation of Conics

The Standard Equation for all Conics

Ax2 + Bxy + Cy2 + Dx + Ey + F = oSo 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 = 0So far B has equal zero and all graphs have

been horizontal or vertical.Not anymore.

How to Find θRotation off the x axisGiven the Conic

BCACot 2

21

2342

0235324 22

Cot

yxyxyx

How to Find θRotation throughGiven the Conic B

CACot 2

21

2342

0235324 22

Cot

yxyxyx

uTanuTan

uCot

uLet

2221

2

1

How to Find θRotation throughGiven the Conic B

CACot 2

21

2342

0235324 22

Cot

yxyxyx

uTanuTan

uCot

uLet

2221

2

1

7.314.632

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°

cossinsincos

yxyyxx

yxyxy

yxyxx

23

2130cos30sin

21

2330sin30cos

Find the coordinates of a point in a rotated coordinate system

Given θ = 60° ; (3, 1) Find

yx ,

yxCosySinxy

yxSinyCosxx

21

236060

23

216060

yx

yx

21

231

23

213

Solve for

System of Equations yx

yx

21

231

23

213 yx ,

yx

yx

yx

36

32

36

Solve for

System of Equations yx

yx

21

231

23

213 yx ,

yx

yx

yx

36

32

36

2331

4362

4362

3362

3632

y

y

yy

yy

Solve for

System of Equations yx

yx

21

231

23

213 yx ,

yx

yx

yx

36

32

36

233

293

212

2936

233136

x

x

x

2331

4362

4362

3362

3632

y

y

yy

yy

Solve for

System of Equations yx

yx

21

231

23

213 yx ,

yx

yx

yx

36

32

36

233

293

212

2936

233136

x

x

x

2331

4362

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

ACBHyperbolaACBParabolaACBCircleorEllipse

Classification of Conics by the Discriminant

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

04 04 04

2

2

2

ACBHyperbolaACBParabolaACBCircleorEllipse

0457612 22 yxyx

Classification of Conics by the Discriminant

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

04 04 04

2

2

2

ACBHyperbolaACBParabolaACBCircleorEllipse

0457612 22 yxyx 3003363671246 2

Ellipse

Finding the intersection of two conics

Given 03

042

22

yxyx

Finding the intersection of two conics

Given 03

042

22

yxyx

1401404)3(

3

2

2

xorxxxxx

yx

Finding the intersection of two conics

Given 03

042

22

yxyx

1

01404)3(

3

2

2

xxxxx

yx

12

432

2

yy

Finding the intersection of two conics

Given 03

042

22

yxyx

1

01404)3(

3

2

2

xxxxx

yx

12

432

2

yy

3

13 2

y

y

Finding the intersection of two conics

Given 03

042

22

yxyx

1

01404)3(

3

2

2

xxxxx

yx

12

432

2

yy

3

13 2

y

y

3,1,3,1

Homework

Page 740 – 741# 3, 19, 29, 47, 52

Page 740 – 741# 5, 23, 35, 50, 55