Question 849052
<pre><font face = "Tohoma" size = 3 color = "indigo"><b> 
Hi,
{{{(y-k)^2/b^2 - (x-h)^2/a^2 = 1}}}
{{{drawing(300,300,   -10,10,-10,12, blue(line(4,10,4,-10)),
 grid(1),
circle(4, 6,0.4),
circle(4, 8,0.4),
circle(4, 4,0.4),
circle(4, 2.84,0.4),
circle(4, 9.16,0.4),
graph( 300, 300, -10,10,-10,12,0,2sqrt((x-4)^2/5 + 1)+6,-2sqrt((x-4)^2/5 +1 )+6   ))}}}
 {{{(y-6)^2/2^2 - (x-4)^2/5 = 1 }}}
C(4,6)  Note: x = 4, 
V(4,8) and V(4,4)
Foci: {{{sqrt(a^2+b^2)}}} = {{{sqrt(10)}}} Foci are: (4, 6+sqrt(10), (4, 6-sqrt(10))

Standard Form of an Equation of an Hyperbola opening up and down is:
  {{{(y-k)^2/b^2 - (x-h)^2/a^2 = 1}}} with C(h,k) and vertices 'b' units up and down from center,  2b the length of the transverse axis
Foci {{{sqrt(a^2+b^2)}}}units units up and down from center, along x = h
& Asymptotes Lines passing thru C(h,k), with slopes m =  ± b/a

Standard Form of an Equation of an Hyperbola opening right and  left is:
  {{{(x-h)^2/a^2 - (y-k)^2/b^2 = 1}}} with C(h,k) and vertices 'a' units right and left of center,   2a the length of the transverse axis
Foci are {{{sqrt(a^2+b^2)}}} units right and left of center along y = k
& Asymptotes Lines passing thru C(h,k), with slopes  m =  ± b/a 
<img src="http://upload.wikimedia.org/wikipedia/commons/thumb/9/97/Hyperbola_properties.svg/200px-Hyperbola_properties.svg.png">