conic sections study guide by david chester types of conic sections circle ellipseparabolahyperbola
TRANSCRIPT
Solving Conics
• Graphing a conic section requires recognizing the type of conic you are given
• To identify the correct form look at key traits of the conic that distinguish it from others
• Once you know what type of conic it is you can start graphing by applying the points and properties starting from the center/vertex
Directory• Formulas
– Circle– Ellipse– Parabola– Hyperbola
• Graphing/Plotting– Circle– Ellipse
• Horizontal• Vertical
– Parabola– Hyperbola
• Horizontal• Vertical
• Differences/Identifying – Circle– Ellipse– Parabola– Hyperbola
Formulas
• Circle:
General Equation for conics:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
(x-h)2 + (y-k)2 = r2
If Center is (0,0):x2 + y2 = r2
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Ellipse Formula
2 2
2 21
x h y k
a b
Axis is horizontal: Axis is Vertical:
2 2
2 21
x h y k
b a
a2 - b2 = c2
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Parabola Formula
• Opens left or right: Opens up or Down:
(y-k)2=4p(x-h) (x-h)2=4p(y-k)
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Hyperbola Formula
• x2 term is positive : y2 is positive:
2 2
2 21
x h y k
a b
2 2
2 21
y k x h
a b
a2 + b2 = c2
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Graphing and Plotting Circles
• Circle:To Graph a Circle:1.Write equation in standard form.2.Place a point for the center (h, k)3.Move “r” units right, left, up and down from center.4.Connect points that are “r” units away from center with smooth curve.
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r
p
Definition of a Circle
A circle is the set of all points in a plane that are equidistant from a fixed point, called the center of the circle. The distance r between the center and any point P on the circle is called the radius.
Differences/Identifying
Discriminant Type of Conic
B2 - 4AC < 0, B = 0, and A = C Circle
B2 - 4AC < 0, and either B does not = 0 or A does not = C
Ellipse
B2 - 4AC = 0 Parabola
B2 - 4AC > 0 Hyperbola
Generally:
Using the General Second Degree Equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 and the properties you can determine the type of conic, more specific ways to identify are on the next few slides.
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Circle Traits
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Examples:•Circles x, y, and r are terms will always be squared or be squares, this does not guarantee perfect squares
•Circles are generally simple formulas as they do not have an a, b, c, or p
2 2 2
2 2
25 144 169
53.29 19.71 73
4 3 5
(7 3) (9 6) 25
Ellipse Traits
• A key point of an ellipse is that you add to equal 1
• In an ellipse a and b term switch with horizontal versus vertical
• a>b• Horizontal: a on the left
side• Vertical: a on right side• a2 - b2 = c2
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Examples:
2 2
2 2
(8 3) ( 5 9)
49 64
(8 3) ( 5 9)
64 49
Parabola Traits
• Parabola is unique because it has a p in its equation
• Only one term is squared
• The x and y switch place with left & right versus up & down
• Up & Down: x on the left• Left & Right: x on the
right
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Examples:
2
2
(6 4) 60(10 5)
(10 5) 60(6 4)
Hyperbola Traits
• A key point for a hyperbola is that you subtract in order to equal 1
• In a hyperbola the x and y terms switch in a horizontal versus a vertical
• Horizontal: x on the left side
• Vertical: x on right side• a2 + b2 = c2
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Examples:
2 2
2 2
(16 7) ( 2 1)
144 169
( 2 1) (16 7)
144 169
Bibliography
• http://math2.org/math/algebra/conics.htm• http://mathforum.org/dr.math/faq/formulas/
faq.analygeom_2.html#twoconicsections• http://www.clausentech.com/lchs/
dclausen/algebra2/formulas/Ch9/Ch9_Conic_Sections_etc_Formulas.doc
• Major Credit to: Kevin Hopp and Sue Atkinson (Slides 9-12 directly from them)