A
CO
NI
CO
PIA
OF
FUN
!
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WHAT ARE THE CONIC SECTIONS?
Menaechmus ~ 350 BC
Archimedes
Apollonius
Pappus ~ 300 AD
Kepler ~ 1620 AD
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THE ELLIPSE
Label center, foci, vertices,
major axis, minor axis.
Eq:
a=
Standard:
b=
General:
c=
e=
V1
P
F1
C
Check definition
Check a2 = b2 + c2
F2
PF1 + PF2 = 2a
error =?
V2
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THE NOT NECESSARILY
FUNCTIONAL PARABOLA
Label focus, vertex, directrix,
Latus Rectum, axis of
symmetry.
Eq:
Directrix:
Axis of symmetry
LR =
Standard:
p=
General:
e=
d
P
V
F
Check LR = 4p
error =?
Check definition
Pd = PF
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THE HYPERBOLA
Eq:
Asymptotes:
Label center, foci, vertices,
transverse axis, conjugate
axis, asymptotes.
a=
Standard:
b=
General:
c=
e=
P
Check c2 = a2 + b2
error =?
Check definition
PF1 - PF2 = 2a
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General form for a conic equation: Ax2 + By2 +Cx +Dy +E = 0
To move the center away from the origin?
Standard form:
Ellipse:
Hyperbola
Parabola
When is a conic equation degenerate?
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Let's complete the square and sketch the conic!
s
49x2 + 9y2 -196x - 54y -164 = 0
k
25x2 - 4y2 + 150x -32y +61 = 0
d
y2 + 8x - 6y + 1 = 0
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Let's write an equation and sketch the graph.
f
p = -3
e=1
vertex at (2,6)
Directrix, y = 9
m
Center at (-2,-3)
Slope of asymptotes: ±1/2
a=1
b=2
c = √5
e=
a
Center at (-2,3)
Vertex at (2,3)
a=4
b =2
c = 2√3
e=
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Can we write an equation for these?
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Examples of conic shapes
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The Human Ellipse
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As the foci move together... a Human Circle
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The Human Hyperbola
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