Find an equation for the hyperbola with vertices 12 and 32 a

Find an equation for the hyperbola with vertices (-1,2) and (3,2) and foci(-2,2) and (4,2). Please include all work. I have the answer already, but a, having a hard time understanding all the steps to get there. Thank you.

Solution

The hyperbola is of the form

(x-h)²/a²- (y-k)²/b²=1

Vertices= (h+-a,k)

Therefore k=2

and h+a=3,h-a=-1

From these equations we get

h=1, a=2

focii=(h+-c,k)

h+c=4   and h-c=-2

h=1, c=3

b^2=c^2 -a^2= 5

Therefore the required equation is

(x-1)²/4 - (y-2)²/5=1