10.5 Hyperbolas

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10.5 Hyperbolas

• L. Kealii Alicea

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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
• 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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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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Standard Form of Hyperbola w/ center _at_ origin
Equation Transverse Axis Asymptotes Vertices
Horizontal y/- (b/a)x (/-a,o)
Vertical y/- (a/b)x (0,/-a)
Foci lie on transverse axis, c units from the
center c2 a2b2
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Graph 4x2 9y2 36

• Write in standard form (divide through by 36)
• a3 b2 because x2 term is transverse axis
  is horizontal vertices are (-3,0) (3,0)
• Draw a rectangle centered at the origin.
• Draw asymptotes.
• Draw hyperbola.

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

• Vertical because foci vertices lie on the
  y-axis
• Center _at_ origin because f v are equidistant
  from the origin
• Since c3 a2, c2 b2 a2
• 9 b2 4
• 5 b2
• /-v5 b

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Assignment10.5A (all)10.5B(odd)