Rotation of Axes; General Form of a Conic

Identifying Conics

Except for degenerate cases:Ax2 + Bxy + Cy2 + Dx + Ey + F = 0(a)Defines a parabola if B2 – 4AC = 0(b)Defines an ellipse (or a circle) if B2 – 4AC < 0(c)Defines a hyperbola if B2 – 4AC > 0

1. Identify the graph of each equation without completing the square(Similar to p.417 #11-20, 43-52)

0762 xyy

2. Identify the graph of each equation without completing the square(Similar to p.417 #11-20, 43-52)

036164 22 yxyx

3. Determine the appropriate rotation formulas to use so that the new

equation contains no xy-term(Similar to p.417 #21-30)

0516 22 yxyx

Rotation Formulas

Standard Form of a Parabola

Standard Form of a Ellipse (a>b)

),(),,(:

),(),,(:

2b :AxisMinor ofLength

2a :AxisMajor ofLength

axis- x toParallel :AxisMajor

),(:

1)()(

22

2

2

2

2

kahkahVertices

kchkchFoci

khCenter

bac

b

ky

a

hx

),(),,(:

),(),,(:

2b :AxisMinor ofLength

2a :AxisMajor ofLength

axis-y toParallel:AxisMajor

),(:

1)()(

22

2

2

2

2

akhakhVertices

ckhckhFoci

khCenter

bac

a

ky

b

hx

Standard Form of a Hyperbola

)(:

),(),,(:

),(),,(:

axis- x toParallel

:Axis Transverse

),(:

1)()(

222

2

2

2

2

hxa

bkyAsymptotes

kahkahVertices

kchkchFoci

khCenter

acb

b

ky

a

hx

)(:

),(),,(:

),(),,(:

axis-y toParallel

:Axis Transverse

),(:

1)()(

222

2

2

2

2

hxb

akyAsymptotes

akhakhVertices

ckhckhFoci

khCenter

acb

b

hx

a

ky

4. Rotate the axes so that the new equation contains no xy-term. Analyze

and graph the new equation(Similar to p.418 #31-42)

4xy

5. Rotate the axes so that the new equation contains no xy-term. Analyze

and graph the new equation(Similar to p.418 #31-42)

0316 22 yxyx

6. Rotate the axes so that the new equation contains no xy-term. Analyze

and graph the new equation(Similar to p.418 #31-42)

01030101069 22 yxyxyx