Find an equation of the hyperbola having foci at (1,-8) and (1,8) and vertices at (1,-5) and (1,5).

To find the equation of the hyperbola, we can start by determining the center of the hyperbola. The center is the midpoint between the two vertices, which in this case is (1, 0).

Next, we can find the distance between the center and one of the foci. The distance between the center (1, 0) and the foci (1, -8) or (1, 8) is 8 units.

The distance between the center and one of the vertices is called the distance "a". In this case, the distance between the center (1, 0) and one of the vertices (1, -5) or (1, 5) is 5 units. Therefore, "a" is equal to 5.

Now, we can find the value of "c", which is the distance between the center and one of the foci. In this case, "c" is equal to 8.

The equation of a hyperbola with a vertical transverse axis is given by:

(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1

Where (h, k) is the center of the hyperbola, "a" is the distance between the center and one of the vertices, and "b" is the distance between the center and one of the co-vertices.

Since the center is (1, 0), "h" is equal to 1 and "k" is equal to 0.

We can substitute the values into the equation:

(x - 1)^2 / 5^2 - (y - 0)^2 / b^2 = 1

Simplifying the equation, we have:

(x - 1)^2 / 25 - (y - 0)^2 / b^2 = 1

Since "c" is equal to 8, we can use the relationship c^2 = a^2 + b^2 to find the value of "b".

8^2 = 5^2 + b^2 64 = 25 + b^2 b^2 = 39

Substituting the value of "b^2" into the equation, we have:

(x - 1)^2 / 25 - (y - 0)^2 / 39 = 1

Therefore, the equation of the hyperbola with the given foci and vertices is:

(x - 1)^2 / 25 - (y - 0)^2 / 39 = 1