Conic Sections Study Guide

By David Chester

Types of Conic Sections

Circle Ellipse Parabola Hyperbola

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.

Graphing and Plotting Ellipses

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Graphing and Plotting Ellipses

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Graphing and Plotting Parabolas

Graphing and Plotting Hyperbolas

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Graphing and Plotting Hyperbolas

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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)