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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
