Equation of the sphere with center `(1,-1,1)` and radius equal to that of sphere
`2x^(2)+2y^(2)+2z^(2)-2x+4y-6z=1` is

Options

A. `x^(2)+y^(2)+z^(2)+2x-2y+2z+1=0`
B. `x^(2)+y^(2)+z^(2)-2x+2y-2z-1=0`
C. `x^(2)+y^(2)+z^(2)+2x+2y-2z+1=0`
D. none of these

Answer

To find the equation of the sphere with center \((1, -1, 1)\) and radius equal to that of the given sphere, we will follow these steps:

### Step 1: Rewrite the given sphere equation
The given equation of the sphere is:
\[ 2x^2 + 2y^2 + 2z^2 - 2x + 4y - 6z = 1 \]
We can divide the entire equation by 2 to simplify it:
\[ x^2 + y^2 + z^2 - x + 2y - 3z = \frac{1}{2} \]

### Step 2: Complete the square for each variable
We will complete the square for \(x\), \(y\), and \(z\).

1. **For \(x\)**:
\[ x^2 - x = \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} \]

2. **For \(y\)**:
\[ y^2 + 2y = (y + 1)^2 - 1 \]

3. **For \(z\)**:
\[ z^2 - 3z = (z - \frac{3}{2})^2 - \frac{9}{4} \]

### Step 3: Substitute back into the equation
Substituting these completed squares back into the equation gives:
\[ \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} + (y + 1)^2 - 1 + \left(z - \frac{3}{2}\right)^2 - \frac{9}{4} = \frac{1}{2} \]

### Step 4: Simplify the equation
Combine the constants on the left side:
\[ \left(x - \frac{1}{2}\right)^2 + (y + 1)^2 + \left(z - \frac{3}{2}\right)^2 - \left(\frac{1}{4} + 1 + \frac{9}{4}\right) = \frac{1}{2} \]
Calculating the constant:
\[ \frac{1}{4} + 1 + \frac{9}{4} = \frac{1 + 4 + 9}{4} = \frac{14}{4} = \frac{7}{2} \]
Thus, we have:
\[ \left(x - \frac{1}{2}\right)^2 + (y + 1)^2 + \left(z - \frac{3}{2}\right)^2 - \frac{7}{2} = \frac{1}{2} \]
Rearranging gives:
\[ \left(x - \frac{1}{2}\right)^2 + (y + 1)^2 + \left(z - \frac{3}{2}\right)^2 = 4 \]

### Step 5: Find the radius
The radius \(r\) of the sphere is \(\sqrt{4} = 2\).

### Step 6: Write the equation of the new sphere
The equation of a sphere with center \((h, k, l)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]
Substituting the center \((1, -1, 1)\) and radius \(2\):
\[ (x - 1)^2 + (y + 1)^2 + (z - 1)^2 = 4 \]

### Final Answer
The equation of the sphere is:
\[ (x - 1)^2 + (y + 1)^2 + (z - 1)^2 = 4 \]