To find the equation of a hyperbola, we need to use the general formula for a hyperbola centered at the origin:
1. Determine the center of the hyperbola: Since the co-vertices are at (4, 0) and (-4, 0), the center is at (0, 0).
2. Calculate the distance from the center to a focus to find 'c': The distance between the center (0, 0) and one of the foci (0, 5) is 5 units. Therefore, 'c' = 5.
3. Calculate 'a' using the relationship between 'a' and 'c': For a hyperbola, a^2 + b^2 = c^2. Since b^2 = 16 (from the given equation), we can find a^2 = c^2 - b^2 = 25 - 16 = 9. Hence, a = 3.
4. Determine the equation based on the values of 'a', 'b', and the center: The standard form of the equation for a hyperbola centered at the origin is:
x^2/a^2 - y^2/b^2 = 1
Therefore, substituting the values of 'a' and 'b':
x^2/9 - y^2/16 = 1
Thus, the equation of the hyperbola is:
x^2/9 - y^2/16 = 1
