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

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

It looks like this:


Therefore the hyperbola has the equation



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 = 

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



We use the Pythagorean relation for hyperbolas to find b:




  <-- what we need for the denominator:



  <--answer



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  in length.
The gold lines are the asymptotes of the hyperbola, the extended diagonals
of the defining rectangle.

Edwin