The equation of the set of points which are equidistant the points (1,-2,3) and (3,-2,-1) is ……….

Answer

To find the equation of the set of points that are equidistant from the points \( A(1, -2, 3) \) and \( B(3, -2, -1) \), we will follow these steps:

### Step 1: Define the Points
Let \( A = (1, -2, 3) \) and \( B = (3, -2, -1) \). Let \( P(x, y, z) \) be any point in space that is equidistant from points \( A \) and \( B \).

### Step 2: Set Up the Distance Equations
The distance from point \( P \) to point \( A \) is given by:
\[ PA = \sqrt{(x - 1)^2 + (y + 2)^2 + (z - 3)^2} \]
The distance from point \( P \) to point \( B \) is given by:
\[ PB = \sqrt{(x - 3)^2 + (y + 2)^2 + (z + 1)^2} \]

### Step 3: Set the Distances Equal
Since \( P \) is equidistant from \( A \) and \( B \), we can set the distances equal to each other:
\[ \sqrt{(x - 1)^2 + (y + 2)^2 + (z - 3)^2} = \sqrt{(x - 3)^2 + (y + 2)^2 + (z + 1)^2} \]

### Step 4: Square Both Sides
To eliminate the square roots, we square both sides:
\[ (x - 1)^2 + (y + 2)^2 + (z - 3)^2 = (x - 3)^2 + (y + 2)^2 + (z + 1)^2 \]

### Step 5: Expand Both Sides
Now, we expand both sides:
- Left Side:
\[ (x - 1)^2 = x^2 - 2x + 1 \]
\[ (y + 2)^2 = y^2 + 4y + 4 \]
\[ (z - 3)^2 = z^2 - 6z + 9 \]
So, the left side becomes:
\[ x^2 - 2x + 1 + y^2 + 4y + 4 + z^2 - 6z + 9 \]
Combining terms gives:
\[ x^2 + y^2 + z^2 - 2x + 4y - 6z + 14 \]

- Right Side:
\[ (x - 3)^2 = x^2 - 6x + 9 \]
\[ (y + 2)^2 = y^2 + 4y + 4 \]
\[ (z + 1)^2 = z^2 + 2z + 1 \]
So, the right side becomes:
\[ x^2 - 6x + 9 + y^2 + 4y + 4 + z^2 + 2z + 1 \]
Combining terms gives:
\[ x^2 + y^2 + z^2 - 6x + 4y + 2z + 14 \]

### Step 6: Set the Expanded Equations Equal
Now we have:
\[ x^2 + y^2 + z^2 - 2x + 4y - 6z + 14 = x^2 + y^2 + z^2 - 6x + 4y + 2z + 14 \]

### Step 7: Cancel Common Terms
Cancel \( x^2, y^2, z^2, \) and \( 14 \) from both sides:
\[ -2x - 6z = -6x + 2z \]

### Step 8: Rearrange the Equation
Rearranging gives:
\[ 6x - 2x - 6z - 2z = 0 \]
This simplifies to:
\[ 4x - 8z = 0 \]

### Step 9: Final Equation
Dividing through by 4 gives:
\[ x - 2z = 0 \]
or
\[ x = 2z \]

### Conclusion
The equation of the set of points that are equidistant from the points \( A(1, -2, 3) \) and \( B(3, -2, -1) \) is:
\[ x - 2z = 0 \]