Final answer:
The equation of the hyperbola with the given properties is (x-10)²/1 - (y-3)²/3 = 1, which does not match any of the provided options.
Step-by-step explanation:
To find the equation of the hyperbola defined by the given properties, we first observe that the foci and vertices are aligned vertically since their x-coordinates are the same. This means the hyperbola opens up and down. With foci at (10,5) and (10,1), the distance between the foci is 4 units. Since foci are 2c units apart, this means c=2. The vertices are (10,3) and (10,1) indicating that the center of the hyperbola is at the midpoint, which is also (10,2). The vertices are 2a units apart, so a=1 because the distance between (10,3) and (10,1) is 2 units. The standard form of a vertical hyperbola centered at (h,k) is (x-h)²/a² - (y-k)²/b² = 1. The value of b can be found using the relationship c² = a² + b², so b² = c² - a² = 4 - 1 = 3. Thus, the equation of the hyperbola in standard form is (x-10)²/1 - (y-3)²/3 = 1, none of the given options matches this result.