Final answer:
The standard form of the equation of the sphere is (x - 2)^2 + y^2 + (z + 5)^2 = 29/2, with the center at (2, 0, -5) and the radius \(\sqrt{29/2}\).
Step-by-step explanation:
The question involves finding the standard form of the equation of a sphere, as well as determining its center and radius. The equation of a sphere in standard form is (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2, where (h, k, l) is the center of the sphere and r is the radius. To convert the given equations into this form, we must complete the square for each of the variables x, y, and z.
First, divide the equation 2x^2 + 2y^2 + 2z^2 = 8x - 20z + 1 by 2 to simplify:
x^2 + y^2 + z^2 = 4x - 10z + \(\frac{1}{2}\)
Now, complete the square for x and z terms:
(x^2 - 4x + 4) + y^2 + (z^2 + 10z + 25) = \(\frac{1}{2}\) + 4 + 25
(x - 2)^2 + y^2 + (z + 5)^2 = \(\frac{29}{2}\)
Therefore, the center of the sphere is at (2, 0, -5) and the radius is \(\sqrt{\frac{29}{2}}\), which is an irrational number.