To rewrite the equation of the sphere in standard form, we start with the given equation:
2x2 + 2y2 + 2z2 – 8x – 24z + 1 = 0
First, we can simplify the equation by dividing everything by 2:
x2 + y2 + z2 – 4x – 12z + rac{1}{2} = 0
Next, we rearrange the equation:
x2 – 4x + y2 + z2 – 12z = - rac{1}{2}
Now we will complete the square for the x and z terms:
For the x terms: x2 – 4x can be rewritten as (x – 2)2 – 4 (since (-4/2)2 = 4).
For the z terms: z2 – 12z can be rewritten as (z – 6)2 – 36 (since (-12/2)2 = 36).
Putting it all together, we have:
(x – 2)2 – 4 + y2 + (z – 6)2 – 36 = - rac{1}{2}
Simplifying gives:
(x – 2)2 + y2 + (z – 6)2 = 39.5
This is now in the standard form of a sphere’s equation: (x – h)2 + (y – k)2 + (z – l)2 = r2, where (h, k, l) is the center and r is the radius.
The center is:
(h, k, l) = (2, 0, 6)
The radius (r) can be calculated as the square root of 39.5:
r = √39.5 ≈ 6.293
In summary:
- Center: (2, 0, 6)
- Radius: ≈ 6.293