Find the equation of the set of points p such that its distance from the points A(3,4,-5) and B(-2,1,4) are equal

Answer

To find the equation of the set of points \( P \) such that its distance from the points \( A(3, 4, -5) \) and \( B(-2, 1, 4) \) are equal, we can follow these steps:

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

### Step 2: Write the Distance Formulas
The distance from point \( P \) to point \( A \) is given by:
\[ PA = \sqrt{(x - 3)^2 + (y - 4)^2 + (z + 5)^2} \]

The distance from point \( P \) to point \( B \) is given by:
\[ PB = \sqrt{(x + 2)^2 + (y - 1)^2 + (z - 4)^2} \]

### Step 3: Set the Distances Equal
Since the distances are equal, we set \( PA = PB \):
\[ \sqrt{(x - 3)^2 + (y - 4)^2 + (z + 5)^2} = \sqrt{(x + 2)^2 + (y - 1)^2 + (z - 4)^2} \]

### Step 4: Square Both Sides
To eliminate the square roots, we square both sides:
\[ (x - 3)^2 + (y - 4)^2 + (z + 5)^2 = (x + 2)^2 + (y - 1)^2 + (z - 4)^2 \]

### Step 5: Expand Both Sides
Now, we expand both sides of the equation:
- Left Side:
\[ (x - 3)^2 = x^2 - 6x + 9 \]
\[ (y - 4)^2 = y^2 - 8y + 16 \]
\[ (z + 5)^2 = z^2 + 10z + 25 \]
So, the left side becomes:
\[ x^2 - 6x + 9 + y^2 - 8y + 16 + z^2 + 10z + 25 = x^2 + y^2 + z^2 - 6x - 8y + 10z + 50 \]

- Right Side:
\[ (x + 2)^2 = x^2 + 4x + 4 \]
\[ (y - 1)^2 = y^2 - 2y + 1 \]
\[ (z - 4)^2 = z^2 - 8z + 16 \]
So, the right side becomes:
\[ x^2 + 4x + 4 + y^2 - 2y + 1 + z^2 - 8z + 16 = x^2 + y^2 + z^2 + 4x - 2y - 8z + 21 \]

### Step 6: Set the Expanded Forms Equal
Now we equate the expanded forms:
\[ x^2 + y^2 + z^2 - 6x - 8y + 10z + 50 = x^2 + y^2 + z^2 + 4x - 2y - 8z + 21 \]

### Step 7: Simplify the Equation
Cancel \( x^2, y^2, z^2 \) from both sides:
\[ -6x - 8y + 10z + 50 = 4x - 2y - 8z + 21 \]

Rearranging gives:
\[ -6x - 4x - 8y + 2y + 10z + 8z + 50 - 21 = 0 \]
\[ -10x - 6y + 18z + 29 = 0 \]

### Step 8: Final Equation
Multiplying through by -1 to simplify:
\[ 10x + 6y - 18z - 29 = 0 \]

This is the equation of the set of points \( P \) such that its distance from points \( A \) and \( B \) are equal.