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
![Page 1: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/1.jpg)
10.5 Rotation of Conics
![Page 2: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/2.jpg)
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.
![Page 3: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/3.jpg)
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.
![Page 4: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/4.jpg)
How to Find θRotation off the x axisGiven the Conic
BCACot 2
21
2342
0235324 22
Cot
yxyxyx
![Page 5: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/5.jpg)
How to Find θRotation throughGiven the Conic B
CACot 2
21
2342
0235324 22
Cot
yxyxyx
uTanuTan
uCot
uLet
2221
2
1
![Page 6: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/6.jpg)
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
![Page 7: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/7.jpg)
When the graph of the conics is moved x and y are called x‘ and y'
Ellipse rotated 45°
yx,
yx ,
![Page 8: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/8.jpg)
Write x and y in the new graph
Let θ = 30°
cossinsincos
yxyyxx
yxyxy
yxyxx
23
2130cos30sin
21
2330sin30cos
![Page 9: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/9.jpg)
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
![Page 10: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/10.jpg)
Solve for
System of Equations yx
yx
21
231
23
213 yx ,
yx
yx
yx
36
32
36
![Page 11: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/11.jpg)
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
![Page 12: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/12.jpg)
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
![Page 13: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/13.jpg)
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
![Page 14: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/14.jpg)
Classification of Conics by the Discriminant
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
04 04 04
2
2
2
ACBHyperbolaACBParabolaACBCircleorEllipse
![Page 15: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/15.jpg)
Classification of Conics by the Discriminant
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
04 04 04
2
2
2
ACBHyperbolaACBParabolaACBCircleorEllipse
0457612 22 yxyx
![Page 16: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/16.jpg)
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
![Page 17: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/17.jpg)
Finding the intersection of two conics
Given 03
042
22
yxyx
![Page 18: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/18.jpg)
Finding the intersection of two conics
Given 03
042
22
yxyx
1401404)3(
3
2
2
xorxxxxx
yx
![Page 19: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/19.jpg)
Finding the intersection of two conics
Given 03
042
22
yxyx
1
01404)3(
3
2
2
xxxxx
yx
12
432
2
yy
![Page 20: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/20.jpg)
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
![Page 21: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/21.jpg)
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
![Page 22: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/22.jpg)
Homework
Page 740 – 741# 3, 19, 29, 47, 52
![Page 23: 10.5 Rotation of Conics](https://reader033.vdocument.in/reader033/viewer/2022061516/56815a8b550346895dc8016d/html5/thumbnails/23.jpg)
Page 740 – 741# 5, 23, 35, 50, 55