If A and B be the points `(3,4,5) and (-1,3,-7)` respectively, find the locus of P such that `PA ^(2) + PB ^(2) = k ^(2).`

Answer

To find the locus of the point \( P \) such that \( PA^2 + PB^2 = k^2 \), where \( A(3, 4, 5) \) and \( B(-1, 3, -7) \), we can follow these steps:

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

### Step 2: Calculate \( PA^2 \) and \( PB^2 \)
The distance \( PA \) from point \( P \) to point \( A \) is given by:
\[ PA^2 = (x - 3)^2 + (y - 4)^2 + (z - 5)^2 \]

The distance \( PB \) from point \( P \) to point \( B \) is given by:
\[ PB^2 = (x + 1)^2 + (y - 3)^2 + (z + 7)^2 \]

### Step 3: Set Up the Equation
According to the problem, we have:
\[ PA^2 + PB^2 = k^2 \]
Substituting the expressions for \( PA^2 \) and \( PB^2 \):
\[ (x - 3)^2 + (y - 4)^2 + (z - 5)^2 + (x + 1)^2 + (y - 3)^2 + (z + 7)^2 = k^2 \]

### Step 4: Expand the Squares
Now, we will expand each term:
1. For \( PA^2 \):
- \( (x - 3)^2 = x^2 - 6x + 9 \)
- \( (y - 4)^2 = y^2 - 8y + 16 \)
- \( (z - 5)^2 = z^2 - 10z + 25 \)

2. For \( PB^2 \):
- \( (x + 1)^2 = x^2 + 2x + 1 \)
- \( (y - 3)^2 = y^2 - 6y + 9 \)
- \( (z + 7)^2 = z^2 + 14z + 49 \)

### Step 5: Combine All Terms
Now we combine all the expanded terms:
\[ (x^2 - 6x + 9) + (y^2 - 8y + 16) + (z^2 - 10z + 25) + (x^2 + 2x + 1) + (y^2 - 6y + 9) + (z^2 + 14z + 49) = k^2 \]

Combining like terms:
- \( 2x^2 \)
- \( 2y^2 \)
- \( 2z^2 \)
- \( -6x + 2x = -4x \)
- \( -8y - 6y = -14y \)
- \( -10z + 14z = 4z \)
- Constant terms: \( 9 + 16 + 25 + 1 + 9 + 49 = 109 \)

Thus, we have:
\[ 2x^2 + 2y^2 + 2z^2 - 4x - 14y + 4z + 109 = k^2 \]

### Step 6: Simplify the Equation
Dividing the entire equation by 2 gives:
\[ x^2 + y^2 + z^2 - 2x - 7y + 2z + \frac{109 - k^2}{2} = 0 \]

### Step 7: Write the Locus Equation
This represents the locus of point \( P \) in the form of a quadratic equation in three dimensions.

### Final Locus Equation
The locus of point \( P \) is given by:
\[ x^2 + y^2 + z^2 - 2x - 7y + 2z = \frac{k^2 - 109}{2} \]