10.5 rotation of conics
DESCRIPTION
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 or vertical. Not anymore. The Standard Equation for all Conics. Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 - PowerPoint PPT PresentationTRANSCRIPT
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