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A hyperbola with one of the vertex at (5,6), the centre 
at (0,6) and one of the slope is equal to 2.


   a) Sketch the graph.
   b) Find the equation.
   c) Locate the foci.
a) Sketch the graph.            y=2x
                       (0,16)
                       B1

      (­5√5, 6)
          F2 A              A1 F1 (5√5, 6)
 a = 5            2
                   C        (5,6)
            (­5,6) (0,6)
 b = 10
 c = 5√5
                       B2
                       (0,­4)
a is the semi­transverse axis. It is the 
          distance from the centre to one of the 
          vertices. 

          b is the semi­conjugate axis. It is 
a = 5
          the distance from the centre to 
b = 10
          one of the points B1 or B2.
c = 5√5

          c is the distance between the centre 
          and one of the foci. It is also equal to 
          the distance between the vertex to one 
          of the points B1 or B2.
We found the distance of the semi­conjugate by 
multiplying the y­coordinate of the vertex to the rise of 
the slope.

           b = 10

We also figured out the distance of the foci by using 
the pythagorean theorem.

              c2 = a2 + b2
b) Find the equation.
            We know from the graph we made 
centre (h,k)
            that the we had a horizontal hyperbola. 
    C (0,6) Therefore the standard equation would
             be:
    a = 5
                           2          2
                      (x­h) _   (y­k) = 1
    b = 10
                        a2         b2
    c = 5√5
By substituting the values the resulting equation 
would be:


                2                2
            (x) _   (y­6) = 1
            25       100
We already figured out where are the locations of the 
foci when we graphed it.

 Given that we know the semi­transverse axis 
 (a) and semi­conjugate axis (b), Use the 
 pythagorean theorem to find (c).
                               2 
                             c = 125
   2     2      2
 c = a  + b
  2   2     2
 c = 5  + 10              c = √125 or 5√5
  2 
 c = 25 + 100
The foci's locations are:


                               F2
   F1
                         (­5√5, 6)
(5√5, 6)

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