Answer
$\frac{(x-1)^{2}}{4} - \frac{(y-2)^{2}}{32} = 1$
Work Step by Step
General Equation of a Hyperbola is: $\frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1$
Foci at (-2, 2) and (4,2); midpoint of the foci is the center, which is at (1,2). h = 1, and k=2
c (distance from focus to center) is equal to 3
Point through (3,2)
Thus so far the equation is $\frac{(x-1)^{2}}{a^{2}} - \frac{(y-2)^{2}}{b^{2}} = 1$
Substitute x=3 and y=2
$\frac{(3-1)^{2}}{a^{2}} - \frac{(2-2)^{2}}{b^{2}} = 1$
$\frac{4}{a^{2}} = 1$
Thus $a^{2} = 4$
$c^{2} = a^{2} + b^{2}$
$6^{2} = 4 + b^{2}$
$b^{2} = 32$
So the final equation is $\frac{(x-1)^{2}}{4} - \frac{(y-2)^{2}}{32} = 1$
