To write the given equation of a sphere in standard form, we need to start with the general equation of a sphere:
(x – h)² + (y – k)² + (z – l)² = r²
Here, (h, k, l) is the center of the sphere, and r is the radius.
The given equation is:
2x² + 2y² + 2z² + 4x + 16z + 1 = 0
First, let’s simplify it by dividing the entire equation by 2:
x² + y² + z² + 2x + 8z + rac{1}{2} = 0
Next, we’ll need to rearrange this into a form that’s easier to work with:
x² + 2x + y² + z² + 8z + rac{1}{2} = 0
Now, we will complete the square for the x and z terms.
For the x terms:
x² + 2x can be rewritten as (x + 1)² – 1.
For the z terms:
z² + 8z can be rewritten as (z + 4)² – 16.
Substituting these back into the equation gives us:
(x + 1)² – 1 + y² + (z + 4)² – 16 + rac{1}{2} = 0
This simplifies to:
(x + 1)² + y² + (z + 4)² – rac{1}{2} – 17 = 0
Combining the constants leads to:
(x + 1)² + y² + (z + 4)² = rac{35}{2}
Now, we can see we have the equation in standard form:
The center of the sphere is at the point (-1, 0, -4).
The radius is √(35/2).
So, summarizing our findings:
- Center: (-1, 0, -4)
- Radius: √(35/2)