10.5 Hyperbolas

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10.5 Hyperbolas
What you should learn
Goal
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Graph and write equations of Hyperbolas.
Goal
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Identify the Vertices and Foci of the hyperbola.
Identify the Foci and Asymptotes.
Goal
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10.5 Hyperbolas
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Hyperbolas

• Like an ellipse but instead of the sum of
  distances it is the difference
• A hyperbola is the set of all points P such that
  the differences from P to two fixed points,
  called foci, is constant

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Hyperbolas

• The line thru the foci intersects the hyperbola
  at two points (the vertices)
• The line segment joining the vertices is the
  transverse axis, and its midpoint is the center
  of the hyperbola.
• Has 2 branches and 2 asymptotes
• The asymptotes contain the diagonals of a
  rectangle centered at the hyperbolas center

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Standard Form of Hyperbola w/ center at origin
Equation Transverse Axis Asymptotes Vertices
Horizontal y /- (b/a)x (/-a,0)
Vertical y /- (a/b)x (0,/-a)
Foci lie on transverse axis, c units from the
center c2 a2b2
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Asymptotes
(0,b)
Vertex (a,0)
Vertex (-a,0)
Focus
Focus
(0,-b)
This is an example of a Horizontal Transverse
axis (a, the biggest number, is under the x2
term with the minus before the y)
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Vertical Transverse axis
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Example) Graph the equation.
36
36
36

• a 3 b 2
• because term is positive, the transverse
  axis is horizontal vertices are
• (-3,0) (3,0)

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Graph 4x2 9y2 36
Example)

• Draw a rectangle centered at the origin.
• Draw asymptotes.
• Draw hyperbola.

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Write the equation of a hyperbola with foci
(0,-3) (0,3) and vertices (0,-2) (0,2).

• Transverse axis is Vertical because foci
  vertices lie on the y-axis
• Center is the origin because foci vertices are
  equidistant from the origin
• Since c 3 a 2, c2 b2 a2
• 9 b2 4
• 5 b2
• /-v5 b

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Reflection on the Section
How are the definitions of ellipse and hyperbola
alike? How are they different?
Both involve all points a certain distance from 2
foci For an ellipse, the sum of the distances
is constant for a hyperbola, the difference is
constant.
assignment
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10.5 Hyperbolas