Final answer:
The equation of the sphere is (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2. We can put the given equation in standard form by completing the square for the x and z terms. The center of the sphere is (2, 0, 6) and the radius is sqrt(-167 - 24z).
Step-by-step explanation:
The equation of the sphere is given by:
(x - h)2 + (y - k)2 + (z - l)2 = r2
In this case, we have the equation 2x2 + 2y2 + 2z2 = 4x - 24z + 1. To put it in standard form, we need to complete the square for the x and z terms. Let's start by completing the square for the x terms:
(x - 2)2 + 2y2 + 2z2 = 4 - 4 - 24z + 1
(x - 2)2 + 2y2 + 2z2 = -23 - 24z
Next, let's complete the square for the z terms:
(x - 2)2 + 2y2 + (z - 6)2 = -23 - 24(6) - 24z
(x - 2)2 + 2y2 + (z - 6)2 = -167 - 24z
Now we can see that the center of the sphere is (2, 0, 6) and the radius is sqrt(-167 - 24z).