10'5 Hyperbolas

1
10.5 Hyperbolas

• p.615

2
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

3
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)
4
Vertical transverse axis
5
Standard Form of Hyperbola w/ center _at_ origin
Foci lie on transverse axis, c units from the
center c2 a2b2
6
Write the equation in standard form
7
Write the equation in standard form
8
Identify the vertices and foci of the hyperbola.
9
Identify the vertices and foci of the hyperbola.
10
Graph the equation. Identify the foci and
asymptotes.
11
Graph the equation. Identify the foci and
asymptotes.
12
Write an equation of the hyperbola with the given
foci and vertices.
13
Write an equation of the hyperbola with the given
foci and vertices.
14
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.

15
(No Transcript)
16
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

17
Worksheet 10.5 B