Question 18281
 what is the equation of a hyperbola with vertex (6,4) and (6,8) and foci(6,0) and (6,12). 

 Mid point of (6,4) & (6,8) is (6,6) : center, and the principal axis is
 vertical.
 
 Assume the equation of the hyperbola is:

{{{ -(x-6)^2/b^2 + (y-6)^2/a^2 = 1 }}}
 Set x = 6, y = 4, we have {{{ 4/a^2 = 1 }}}, so {{{ a^2 = 4 }}}
 (or use dist|(6,4),(6,12)| - dist|(6,4),(6,0)| = 4 = 2a to get a = 2)

 Since the distance between two foci (6,0) & (6,12) is 12 = 2c, so c = 6.
 by  c^2 = a^2 + b^2 = 36,we have b^2 = 32. [Note e = c/a = 3]
  
 Therefore, the equation is 
 {{{ -(x-6)^2/32 + (y-6)^2/4 = 1 }}}


 Kenny