The Hyperbola

1
Section 8-3

• The Hyperbola

2
Section 8-3

• the geometric definition of a hyperbola
• standard form of a hyperbola with a center at
  (0 , 0)
• translating a hyperbola center at (h , k)
• graphing a hyperbola
• finding the equations of the asymptotes
• finding the equation of a hyperbola
• eccentricity and orbits
• reflective properties of hyperbola

3
Geometry of a Hyperbola

• hyperbola the set of all points whose
  distance from two fixed points (the foci) have
  a constant difference
• all the points are coplanar
• the line through the foci is called the focal
  axis
• the midway point between the foci is called the
  center

4
Geometry of a Hyperbola
F1
F2
V1
V2
center
F1 and F2 are the foci V1 and V2 are the vertices
(chord between called the transverse axis)
5
Geometry of a Hyperbola
F1
F2
V1
V2
center
d2
d1
F1 and F2 are the foci d1 - d2
constant V1 and V2 are the vertices
(chord between called the transverse axis)
6
Standard Form Center (0 , 0)

• 2a length of the transverse axis (endpoints
  are the vertices)
• 2b length of the conjugate axis
• c focal radius (distance from the center to
  each foci)
• c2 a2 b2 (use to find c)

7
Standard Form Center (h , k)
8
Graphing a Hyperbola

• convert the equation into standard form, if
  necessary (complete the square)
• find and plot the center
• use a to plot the vertices (same direction as
  the variable a2 is underneath)
• use b to plot two other points
• draw a rectangle using these four points
• draw the diagonals of the rectangle (dashed),
  these are the asymptotes
• draw in the hyperbola (use vertices)
• plot the foci using c (c is the distance from
  the center to each focus)

9
Equations of the Asymptotes

• the equations of the asymptotes can be found by
  replacing the 1 on the right-side of the equation
  with a 0 and then solving for y