Find the equation of the set of points P such that `PA^(2)+PB^(2)=2k^(2)` where A,B are the points (3,4,5) (-1,3,-7) respectively

Answer

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

### Step 1: Define the Points
Let \( P \) be the point \( (x, y, z) \). The points \( A \) and \( B \) are given as:
- \( A(3, 4, 5) \)
- \( B(-1, 3, -7) \)

### Step 2: Write the Distance Formulas
Using the distance formula, we can express \( PA^2 \) and \( PB^2 \):
\[ PA^2 = (x - 3)^2 + (y - 4)^2 + (z - 5)^2 \]
\[ PB^2 = (x + 1)^2 + (y - 3)^2 + (z + 7)^2 \]

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

### Step 4: 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. \( (z - 5)^2 = z^2 - 10z + 25 \)
4. \( (x + 1)^2 = x^2 + 2x + 1 \)
5. \( (y - 3)^2 = y^2 - 6y + 9 \)
6. \( (z + 7)^2 = z^2 + 14z + 49 \)

Combining these, we get:
\[ (x^2 - 6x + 9) + (y^2 - 8y + 16) + (z^2 - 10z + 25) + (x^2 + 2x + 1) + (y^2 - 6y + 9) + (z^2 + 14z + 49) \]

### Step 5: Combine Like Terms
Now, we combine like terms:
- For \( x^2 \): \( 2x^2 \)
- For \( y^2 \): \( 2y^2 \)
- For \( z^2 \): \( 2z^2 \)
- For \( x \): \( -6x + 2x = -4x \)
- For \( y \): \( -8y - 6y = -14y \)
- For \( z \): \( -10z + 14z = 4z \)

Now, summing the constant terms:
\[ 9 + 16 + 25 + 1 + 9 + 49 = 109 \]

### Step 6: Write the Final Equation
Putting everything together, we have:
\[ 2x^2 + 2y^2 + 2z^2 - 4x - 14y + 4z = 2k^2 - 109 \]
This can be rearranged to:
\[ 2x^2 + 2y^2 + 2z^2 - 4x - 14y + 4z + 109 - 2k^2 = 0 \]

### Final Result
Thus, the equation of the set of points \( P \) is:
\[ 2x^2 + 2y^2 + 2z^2 - 4x - 14y + 4z = 2k^2 - 109 \]