Final answer:
The equation of the sphere in standard form is (x - 2)^2 + (y - 2/3)^2 + z^2 = 52/9, with the center at (2, 2/3, 0) and radius (2√13)/3.
Step-by-step explanation:
The equation of the sphere in the question is 9x2 + 9y2 + 9z2 = 36x - 12y - 24.
To write it in standard form, we need to complete the square for each variable and move all the constant terms to the other side of the equation.
First, divide the entire equation by 9 to simplify it:
x2 + y2 + z2 = 4x - (4/3)y - (8/3)
Now, complete the square for each of the variables x, y, and z:
(x - 2)2 - 4 + (y - 2/3)2 - 4/9 + (z2) = 0
Rearrange the terms to form the equation of the sphere in standard form:
(x - 2)2 + (y - 2/3)2 + z2 = (8/3) + 4 + 4/9 = 52/9
The center of the sphere is located at (2, 2/3, 0) and the radius is the square root of 52/9, which simplifies to (2√13)/3.