10'3 Hyperbolas

1
10.3 Hyperbolas

• A hyperbola is the set of all points, whose
  distances from two fixed points (foci)
• differ by a positive constant.
• x2/a2 - y2/b2 1
  or y2/a2 - x2/b2 1
• Where the center is the origin
• the transverse (intercepted) axis has length of
  2a
• the conjugate (untouched) axis has length of 2b
• The focal points are located on the transverse
  axis a distance c units from the center.
• The value of c is calculated using the equation
  c2 a2 b2, which ensures that the focal points
• will exist beyond the vertices of the hyperbola.
• Sketching the graph of hyperbolas identify
  transverse axis by finding intercepts
• mark length of the conjugate axis
• sketch asymptotes as diagonals of auxiliary
  rectangle

2

• If y is the transverse axis
• y2/a2 - x2/b2 1
• y-intercepts y2/a2 1, y2 a2, y a
• x-intercepts -x2/b2 1, x2 -b2, x imaginary
  number
• The x-intercepts do not exist. However, the
  conjugate axis still marks a length of b units
  right and left.
• The asymptote equations would be derived from y -
  y1 m (x-x1). In this case, y a/b x.
• y2 - 4x2 36, y2/36 - x2/9 1
• y-intercepts y2/36 1, y2 36, y 6
• x-intercepts -x2/9 1, x2 -9, no intercepts