Write an equation for the hyperbola that satisfies
each set of conditions.                  __                   
vertices (9,-3) and (-5,-3), foci at (2±Ö53,-3)

This is a hyperbola that looks like this:   }{

and has equation

(x-h)²   (y-k)² 
—————— - —————— = 1
  a²       b² 

where center = (h,k),
a = semi-transverse axis, 
b = semi conjugate axis,
foci (h±c,k) where c² = a²+b²,
and asaymptotes y - k = ±(b/a)(x - h)

The transverse axis is the line that goes from one vertex to the other.
The distance between (9,-3) and (-5,-3) is 14 units.
Therefore the semi-transverse axis, a = 7 units.

So far we have:

(x-h)²   (y-k)² 
—————— - —————— = 1
  7²       b² 

The center is midway between the foci, and since the midpoint between
(9,-3) and (-5,-3) is (2,-3), that is the center (h,k)
h = 2, k = -3

So far we have:

(x-2)²   (y+3)² 
—————— - —————— = 1
  7²       b² 

We only need b.
                          __
and we are given that (2±Ö53,-3) are the foci.
We know that the foci = (h±c,k), or (2±c,-3), so
c = Ö53

Also we know that
                               __
c² = a²+b², and a = 7 and c = Ö53
  __
(Ö53)² = 7² + b²

  53 = 49 + b²
   4 = b²
   b = 2

So now we have b and the hyperbola is

(x-2)²   (y+3)² 
—————— - —————— = 1
  7²       2²

or


(x-2)²   (y+3)² 
—————— - —————— = 1
  49       4


      
     hyperbola only                   with asymptotes 

Edwin
AnlytcPhil@aol.com