10.5 rotation of conics. the standard equation for all conics ax 2 + bxy + cy 2 + dx + ey + f = o so...
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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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# 3, 19, 29, 47, 52
Page 740 – 741
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