Conics: a crash course
MathScience Innovation Center
Betsey Davis
Why “conics”?
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The 4 basic shapes are formed by slicing
a right circular cone
What is a right circular cone?
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A cone, with a circular base, whose axis is
perpendicular to that base.
Conics
B. Davis
MathScience Innovation Center
Not right circular cone:
Conics
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MathScience Innovation Center
What are the 4 basic conics?
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Parabola
Circle
Ellipse
Hyperbola
Conics
B. Davis
MathScience Innovation Center
What is the relationship between
the cone and the 4 shapes?

It’s how you slice !
Conics
B. Davis
MathScience Innovation Center
Slicing a cone
Take notes on first site!
You will be responsible for
knowing some real-world
applications of each of the
conics.
Let’s visit
1http://id.mind.net/~zona/mmts/miscellaneousMat
h/conicSections/conicSections.htm
2http://ccins.camosun.bc.ca/~jbritton/jbconics.htm
3http://www.keypress.com/sketchpad/java_gsp/co
nics.html
4http://www.exploremath.com/activities/activity_lis
t.cfm?categoryID=1
Conics
B. Davis
MathScience Innovation Center
General Equation:

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
•For us,
B = 0 always
(this rotates the conic between 0 and 90 degrees)
Conics
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MathScience Innovation Center
General Equation:

Ax^2 + By^2 + Cx + Dy + E = 0
•What is the value of A or B if it is a
parabola?
•B=0 or A =0 but not both
Conics
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MathScience Innovation Center
General Equation:

Ax^2 + By^2 + Cx + Dy + E = 0
•If circle
•B=A
Conics
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MathScience Innovation Center
General Equation:

Ax^2 + By^2 + Cx + Dy + E = 0
•If ellipse
•B is not equal to A, but they have the
same sign
Conics
B. Davis
MathScience Innovation Center
General Equation:

Ax^2 + By^2 + Cx + Dy + E = 0
•If hyperbola
•B and A have opposite signs
Conics
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MathScience Innovation Center

General Equation:

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3x^2 + 3y^2 + 2x + y + 8 = 0
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3x^2 - 3y^2 + 2x + y + 8 = 0
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Parabola
Circle
Ellipse
Hyperbola
3x^2 + 9y^2 + 2x + y + 8 = 0

3x^2 + 2x + y + 8 = 0
Conics
B. Davis
MathScience Innovation Center
Parabola Reminders
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Parabolas opening up and down are the
only conics that are functions
Y = (x-3)^2 +4
Vertex? (3,4)
Axis of symmetry? X = 3
Opening which way?
up
Conics
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MathScience Innovation Center
Parabola Reminders
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Y^2 –4Y + 3 –x = 0
Vertex? (-1,2)
Axis of symmetry? Y=2
Opening which way? right
Conics
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MathScience Innovation Center
Circles
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Ax^2 + Ay^2 +Cx + Dy + E= 0
(x-h)^2 + (y-K)^2 = r^2
Where (h,k) is the center and r is the
radius
•X^2 + y^2 = 36
•Centered at origin
•Radius is 6
Conics
B. Davis
MathScience Innovation Center
Circles


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Ax^2 + Ay^2 +Cx + Dy + E= 0
(x-h)^2 + (y-K)^2 = r^2
Where (h,k) is the center and r is the
radius
•(X-1)^2 +( y-3)^2 = 49
•Center at (1,3)
•Radius is 7
Conics
B. Davis
MathScience Innovation Center
Ellipses
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Ax^2 + By^2 +Cx + Dy + E= 0
(x-h)^2 + (y-K)^2 = 1
a^2
b^2
Where (h,k) is the center and a is the long
radius and b is the short radius
•(X)^2 +( y)^2
25
=1
4
•Center at (0,0)
•Major axis 10, minor 4
Conics
B. Davis
MathScience Innovation Center
Ellipses
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
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Ax^2 + By^2 +Cx + Dy + E= 0
(x-h)^2 + (y-K)^2 = 1
a^2
b^2
Where (h,k) is the center and a is the long
radius and b is the short radius
•(X-1)^2 +( y+3)^2
16
=1
100
•Center at (1,-3)
•Major axis 20, minor 8
Conics
B. Davis
MathScience Innovation Center
Hyperbolas
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Ax^2 - By^2 +Cx + Dy + E= 0
(x-h)^2 - (y-K)^2 = 1
a^2
b^2
Where (h,k) is the center and 2a is the
transverse axis
•(X-1)^2 -( y+3)^2
16
=1
100
•Center at (1,-3)
•Transverse axis length is 8
Conics
B. Davis
MathScience Innovation Center
Hyperbolas



Ax^2 - By^2 +Cx + Dy + E= 0
(x-h)^2 - (y-K)^2 = 1
a^2
b^2
Where (h,k) is the center and 2a is the
transverse axis
•(y)^2
16
-
( x)^2
=1
100
•Center at (0,0)
•Transverse axis length is 8
Conics
B. Davis
MathScience Innovation Center