Conic Sections

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Conic Sections

• Hyperbolas

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Definition

• A hyperbola is the set of all points in the plane
  in which the difference of the distances from two
  distinct fixed points, called the foci is
  constant.
• The center is the midpoint of the line segment
  whose endpoints are the foci.

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Definition

• A hyperbola has TWO axis of symmetryThe
  transverse axis (distance between the vertices)
  2aThe conjugate axis (perpendicular to the
  transverse axis and through the center)

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Definition

• A hyperbola also has two asymptotes that describe
  the end behavior of the curves.(Well look at
  those equations tomorrow!)

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The Equations

• General equation

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Standard Equation
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Standard Equation
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Example 1

• Find the equation of the hyperbola with foci at
  (7, 1) and (-3 ,1) whose transverse axis is 8
  units long.
• Does the hyperbola open up and down or right and
  left?
• Find the center Use the midpoint formula

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Example 1

• Find the equation of the hyperbola with foci at
  (7, 1) and (-3 ,1) whose transverse axis is 8
  units long.
• Now Ive got (h, k) and all I need is a and b.
  a is the distance from the center to the
  vertexc is the distance from the center to the
  focib2 c2 - a2

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Example 1

• Find the equation of the hyperbola with foci at
  (7, 1) and (-3 ,1) whose transverse axis is 8
  units long.
• The transverse axis is 8 units long, so the
  distance from the center to a vertex is 4. (a4)
• The distance from the center (2, 1) to a focus
  (7,1) 5 (c5)

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Example 1

• Find the equation of the hyperbola with foci at
  (7, 1) and (-3 ,1) whose transverse axis is 8
  units long.

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Example 2

• Find the coordinates of the center, foci and
  vertices of the graph of
• Does the hyperbola open up and down or right and
  left?
• Find the center (2, -4)
• Vertices are (h, k?a) (2, -10) and (2, 2)

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Example 2

• Find the coordinates of the center, foci and
  vertices of the graph of
• The foci are (h, k?c) and a2 b2 c2

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Example 3 (the hard one)

• Find the coordinates of the center, foci and
  vertices of the graph of
• Rearrange and group
• Factor GCF for each variable
• Complete the square (you try it)

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Example 3 (the hard one)

• Find the coordinates of the center, foci and
  vertices of the graph of

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Example 3 (the hard one)

• Find the coordinates of the center, foci and
  vertices of the graph of
• Find a, b c
• Center
• Foci
• Vertices

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Homework

• Page 720 in packet
• 1-4 all
• 5-13 odd (no asymptotes no graphing)
• 19-21 odd (no asymptotes graph on grapher)