Question 1206860
<pre>
The vertices and foci lie on the horizontal line y=2, since all their y-coordinates have y-coordinate 2.

It looks like this:

{{{drawing(400,320,-7,3,-2,6,
circle(-4,2,.05), circle(0,2,.05),circle(-5,2,.05),circle(1,2,.05),circle(-2,2,.05),
graph(400,320,-7,3,-2,6,(1/2)(4 - sqrt(5)*sqrt(x^2 + 4x))),
 
graph(400,320,-7,3,-2,6,(1/2)(4 + sqrt(5)*sqrt(x^2 + 4x)))

)}}}
Therefore the hyperbola has the equation

{{{(x-h)^2/a^2-(y-k)^2/b^2}}}{{{""=""}}}{{{1}}}

where the vertex is the midpoint between vertices, and also the midpoint
between foci. That is, the vertex is (-2,2).

a = semi-transverse axis = distance from center to vertex = 2 units
c = semi-conjugate axis = half the height of defining rectangle = {{{sqrt(5)}}}

Find the equation of the hyperbola with vertices at (-4,2) 
and (0,2) and foci at (-5,2) and (1,2).

So we have the center, so we can determine everything about the equation
except b.

(h,k) the center = (-2,2), a=2

{{{(x+2)^2/2^2-(y-2)^2/b^2}}}{{{""=""}}}{{{1}}}

We use the Pythagorean relation for hyperbolas to find b:

{{{c^2=a^2+b^2}}}
{{{3^2=2^2+b^2}}}
{{{9=4+b^2}}}
{{{5=b^2}}}  <-- what we need for the denominator:
{{{sqrt(5)=b}}}

{{{(x+2)^2/2^2-(y-2)^2/5^""}}}{{{""=""}}}{{{1}}}
{{{(x+2)^2/4-(y-2)^2/5^""}}}{{{""=""}}}{{{1}}}  <--answer

{{{drawing(400,320,-7,3,-2,6,
circle(-4,2,.05), circle(0,2,.05),circle(-5,2,.05),circle(1,2,.05),circle(-2,2,.05),
graph(400,320,-7,3,-2,6,(1/2)(4 - sqrt(5)*sqrt(x^2 + 4x))),
 
graph(400,320,-7,3,-2,6,(1/2)(4 + sqrt(5)*sqrt(x^2 + 4x))),
red(line(-2,2-sqrt(5),-2,2+sqrt(5))),

graph(400,320,-7,3,-2,6,20,25,21,15,1.118033989x+4.2360679775),
graph(400,320,-7,3,-2,6,20,25,21,15,-1.118033989x-.2360679775),
blue(line(-4,2,0,2)),
green(line(-4,2-sqrt(5),0,2-sqrt(5)), line(0,2-sqrt(5),0,0+sqrt(5)),line(-4,2+sqrt(5),0,2+sqrt(5)),line(-4,2+sqrt(5),0,2+sqrt(5)),line(-4,2-sqrt(5),0,2-sqrt(5)), line(-4,2-sqrt(5),-4,2+sqrt(5))))}}}

The defining rectangle is in green.
The blue line is the transverse axis, 2a or 4 in length
The red line is the conjugate axis 2b or {{{2sqrt(5)}}} in length.
The gold lines are the asymptotes of the hyperbola, the extended diagonals
of the defining rectangle.

Edwin</pre>