Hyberbola

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Hyberbola

• Conic Sections

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Hyperbola

• The plane can intersect two nappes of the cone
  resulting in a hyperbola.

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Hyperbola - Definition
A hyperbola is the set of all points in a plane
such that the difference in the distances from
two points (foci) is constant.
d1 d2 is a constant value.
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Finding An Equation

• Hyperbola

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Hyperbola - Definition
What is the constant value for the difference in
the distance from the two foci? Let the two foci
be (c, 0) and (-c, 0). The vertices are (a, 0)
and (-a, 0).
d1 d2 is the constant.
If the length of d2 is subtracted from the left
side of d1, what is the length which remains?
d1 d2 2a
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Hyperbola - Equation
Find the equation by setting the difference in
the distance from the two foci equal to 2a.
d1 d2 2a
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Hyperbola - Equation
Simplify
Remove the absolute value by using or -.
Get one square root by itself and square both
sides.
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Hyperbola - Equation
Subtract y2 and square the binomials.
Solve for the square root and square both sides.
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Hyperbola - Equation
Square the binomials and simplify.
Get xs and ys together on one side.
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Hyperbola - Equation
Factor.
Divide both sides by a2(c2 a2)
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Hyperbola - Equation
Let b2 c2 a2
where c2 a2 b2
If the graph is shifted over h units and up k
units, the equation of the hyperbola is
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Hyperbola - Equation
where c2 a2 b2
RecognitionHow do you tell a hyperbola from an
ellipse?
AnswerA hyperbola has a minus (-) between the
terms while an ellipse has a plus ().
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Graph - Example 1

• Hyperbola

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Hyperbola - Graph
Graph
Center
(-3, -2)
The hyperbola opens in the x direction because
x is positive.
Transverse Axis
y -2
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Hyperbola - Graph
Graph
Vertices
(2, -2) (-4, -2)
Construct a rectangle by moving 4 units up and
down from the vertices.
Construct the diagonals of the rectangle.
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Hyperbola - Graph
Graph
Draw the hyperbola touching the vertices and
approaching the asymptotes.
Where are the foci?
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Hyperbola - Graph
Graph
The foci are 5 units from the center on the
transverse axis.
Foci (-6, -2) (4, -2)
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Hyperbola - Graph
Graph
Find the equation of the asymptote lines.
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3
Use point-slope formy y1 m(x x1) since
the center is on both lines.
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Slope
Asymptote Equations
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Graph - Example 2

• Hyperbola

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Hyperbola - Graph
Sketch the graph without a grapher
RecognitionHow do you determine the type of
conic section?
AnswerThe squared terms have opposite signs.
Write the equation in hyperbolic form.
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Hyperbola - Graph
Sketch the graph without a grapher
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Hyperbola - Graph
Sketch the graph without a grapher
Center
(-1, 2)
Transverse Axis Direction
Up/Down
Equation
x-1
Vertices
Up/Down from the center or
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Hyperbola - Graph
Sketch the graph without a grapher
Plot the rectangular points and draw the
asymptotes.
Sketch the hyperbola.
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Hyperbola - Graph
Sketch the graph without a grapher
Plot the foci.
Foci
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Hyperbola - Graph
Sketch the graph without a grapher
Equation of the asymptotes
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Finding an Equation

• Hyperbola

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Hyperbola Find an Equation
Find the equation of a hyperbola with foci at (2,
6) and (2, -4). The transverse axis length is 6.
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Conic Section Recogition
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Recognizing a Conic Section
Parabola -
One squared term. Solve for the term which is
not squared. Complete the square on the squared
term.
Ellipse -
Two squared terms. Both terms are the same
sign.
Circle -
Two squared terms with the same coefficient.
Hyperbola -
Two squared terms with opposite signs.