To find the center and radius of the sphere from the given equation, we can rewrite the equation in standard form. The given equation is:
x² + y² + z² – 4x – 2z – 8 = 0
We can rearrange this equation to isolate the variables:
x² – 4x + y² + z² – 2z = 8
Now, we complete the square for the x and z terms:
1. For the x terms: x² – 4x can be completed as follows:
Take half of -4, which is -2, and square it to get 4. So, we write:
(x – 2)² – 4
2. For the z terms: z² – 2z can be completed similarly:
Take half of -2, which is -1, and square it to get 1. So, we write:
(z – 1)² – 1
Substituting these completed squares back into the equation, we have:
(x – 2)² – 4 + y² + (z – 1)² – 1 = 8
Simplifying this gives:
(x – 2)² + y² + (z – 1)² – 5 = 8
Which leads us to:
(x – 2)² + y² + (z – 1)² = 13
From the standard form of a sphere’s equation:
(x – h)² + (y – k)² + (z – l)² = r²
We can identify the center (h, k, l) and the radius r:
The center is at: (2, 0, 1)
The radius is: √13
Thus, the center of the sphere is (2, 0, 1) and the radius is √13.