Final answer:
The equation of the hyperbola with the given center, vertices, and foci is determined using the relationships between the lengths a, b, and c. Upon calculations, the standard equation of this hyperbola with a horizontal orientation is found to be (x - 2)²/9 - (y - 2)²/16 = 1. However, this doesn't match any of the provided options.
Step-by-step explanation:
To find the equation of the hyperbola, you need to know the center, the distance between the center and the vertices (a), and the distance between the center and the foci (c). In this case, the hyperbola has a horizontal orientation because the vertices and foci have the same y-coordinates but different x-coordinates.
The center of the hyperbola is given as (2, 2). The distance between the center (2, 2) and the vertices at (-1, 2) and (5,2) is 3 units, so a = 3. This means a^2 = 9. The distance from the center to the foci at (-3, 2) and (7, 2) is 5 units, so c = 5, and hence c^2 = 25.
For a hyperbola, c^2 = a^2 + b^2. By substituting the known values, we get 25 = 9 + b^2, resulting in b^2 = 16.
Therefore, the equation of the hyperbola is (x - 2)²/9 - (y - 2)²/16 = 1, which matches none of the given options. There may be an error in the provided options or in the interpretation of the question.