Find the equation of the hyperbola that has a center at (8,1), a focus at (3,1), and a vertex at (4,1).

To find the equation of a hyperbola, we need to know the center, the foci, and the vertices. In this case, the center is (8,1), the focus is (3,1), and the vertex is (4,1).

Step 1: Determine the distance between the center and the focus, which is the distance between (8,1) and (3,1). The distance formula is:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Substituting the coordinates, we have:
Distance = sqrt((3 - 8)^2 + (1 - 1)^2)
= sqrt((-5)^2 + 0^2)
= sqrt(25)
= 5

Step 2: Determine the distance between the center and the vertex, which is the distance between (8,1) and (4,1). Using the distance formula again:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Substituting the coordinates, we have:
Distance = sqrt((4 - 8)^2 + (1 - 1)^2)
= sqrt((-4)^2 + 0^2)
= sqrt(16)
= 4

Step 3: Determine if the hyperbola is horizontal or vertical. Since the y-coordinates of the center, focus, and vertex are the same, the hyperbola is horizontal.

Step 4: Write the equation in standard form. For a horizontal hyperbola, the standard form of the equation is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1

Where (h,k) is the center of the hyperbola and "a" and "b" are the distances from the center to the vertices and foci, respectively.

Substituting the given values, we have:
(x - 8)^2 / 4^2 - (y - 1)^2 / 5^2 = 1

Simplifying, we get the equation of the hyperbola:
(x - 8)^2 / 16 - (y - 1)^2 / 25 = 1