Final answer:
The standard form of the equation of the hyperbola with the given conditions is x² / 1 - y² / 99 = 1.
Step-by-step explanation:
The standard form of the equation of a hyperbola can be written as:
(x-h)² / a² - (y-k)² / b² = 1
To find the standard form, we need to determine the values of h, k, a, and b.
Given that the foci are at (-5,0) and (5,0), the center of the hyperbola is at the origin (0,0). So, h = 0 and k = 0.
Next, we can find the values of a and b using the distance formula
The distance between the center and each vertex is equal to a. In this case, the distance between the center (0,0) and the vertex (1,0) is 1. So, a = 1.
The distance between the center and each focus is equal to c, where c is the distance between the foci. In this case, the distance between the foci (-5,0) and (5,0) is 10. So, c = 10.
Using the relationship c² = a² + b², we can solve for b:
10² = 1² + b²
b² = 100 - 1 = 99
b = √99
Therefore, the standard form of the equation of the hyperbola is:
x² / 1 - y² / 99 = 1