Final answer:
To find the equation of the hyperboloid of one sheet passing through the given points, we need to determine the values of a, b, and c.
Step-by-step explanation:
To find the equation of the hyperboloid of one sheet passing through the given points, we need to determine the equation in the form x^2/a^2 - y^2/b^2 - z^2/c^2 = 1. Let's start by finding the values of a, b, and c. Using the points (±5,0,0) and (0,±8,0), we can see that a = 5 and b = 8.
Substituting these values into the equation, we get: x^2/5^2 - y^2/8^2 - z^2/c^2 = 1. Now, let's use the points (±10,0,4) and (0,±16,4) to find the value of c. By substituting the values into the equation, we can solve for c. After substituting the values into the equation, we get: (10^2/5^2) - (16^2/8^2) - (4^2/c^2) = 1.
By solving this equation, we find c = ±6. Therefore, the equation of the hyperboloid of one sheet passing through the given points is: x^2/25 - y^2/64 - z^2/36 = 1.