If A(3,4,1) and B(-1,2,3) are two points, then find the locus of a moving point P such that `PA^(2)+PB^(2)=2k^(2)`.

Explanation

To find the locus of a moving point \( P \) such that \( PA^2 + PB^2 = 2k^2 \), where \( A(3,4,1) \) and \( B(-1,2,3) \), we will follow these steps:

### Step 1: Define the coordinates of point \( P \)
Let the coordinates of point \( P \) be \( (x, y, z) \).

### Step 2: Calculate \( PA^2 \)
Using the distance formula, the square of the distance \( PA \) from point \( P \) to point \( A \) is given by:
\[ PA^2 = (x - 3)^2 + (y - 4)^2 + (z - 1)^2 \]

### Step 3: Calculate \( PB^2 \)
Similarly, the square of the distance \( PB \) from point \( P \) to point \( B \) is:
\[ PB^2 = (x + 1)^2 + (y - 2)^2 + (z - 3)^2 \]

### Step 4: Set up the equation
According to the problem, we have:
\[ PA^2 + PB^2 = 2k^2 \]
Substituting the expressions for \( PA^2 \) and \( PB^2 \):
\[ (x - 3)^2 + (y - 4)^2 + (z - 1)^2 + (x + 1)^2 + (y - 2)^2 + (z - 3)^2 = 2k^2 \]

### Step 5: Expand the squares
Now we will expand each term:
1. \( (x - 3)^2 = x^2 - 6x + 9 \)
2. \( (y - 4)^2 = y^2 - 8y + 16 \)
3. \( (z - 1)^2 = z^2 - 2z + 1 \)
4. \( (x + 1)^2 = x^2 + 2x + 1 \)
5. \( (y - 2)^2 = y^2 - 4y + 4 \)
6. \( (z - 3)^2 = z^2 - 6z + 9 \)

### Step 6: Combine all terms
Combining all the expanded terms:
\[ (x^2 - 6x + 9) + (y^2 - 8y + 16) + (z^2 - 2z + 1) + (x^2 + 2x + 1) + (y^2 - 4y + 4) + (z^2 - 6z + 9) = 2k^2 \]

This simplifies to:
\[ 2x^2 + 2y^2 + 2z^2 - 4x - 12y - 8z + 40 = 2k^2 \]

### Step 7: Simplify the equation
Dividing the entire equation by 2 gives:
\[ x^2 + y^2 + z^2 - 2x - 6y - 4z + 20 = k^2 \]

### Conclusion
The locus of the point \( P \) is given by the equation:
\[ x^2 + y^2 + z^2 - 2x - 6y - 4z + 20 = k^2 \]