Find the equation of the set of the points `P` such that is distances from the points `A(3,4,-5)` and `B(-1,2,4)` are equal.

Options

A. `8x+y-18z-29=0`
B. `8x+4y-18z-29=0`
C. `x+4y-18z-29=0`
D. `8x-4y-18z-29=0`

Answer

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

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

### Step 2: Write the distance equations
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:
\[ PB = \sqrt{(x + 1)^2 + (y - 2)^2 + (z - 4)^2} \]

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

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

### Step 5: Expand both sides
Expanding the left side:
\[ (x^2 - 6x + 9) + (y^2 - 8y + 16) + (z^2 + 10z + 25) \]
This simplifies to:
\[ x^2 + y^2 + z^2 - 6x - 8y + 10z + 50 \]

Expanding the right side:
\[ (x^2 + 2x + 1) + (y^2 - 4y + 4) + (z^2 - 8z + 16) \]
This simplifies to:
\[ x^2 + y^2 + z^2 + 2x - 4y - 8z + 21 \]

### Step 6: Set the expanded equations equal
Now we have:
\[ x^2 + y^2 + z^2 - 6x - 8y + 10z + 50 = x^2 + y^2 + z^2 + 2x - 4y - 8z + 21 \]

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

### Step 8: Rearrange the equation
Now, rearranging gives:
\[ -6x - 2x - 8y + 4y + 10z + 8z + 50 - 21 = 0 \]
This simplifies to:
\[ -8x - 4y + 18z + 29 = 0 \]

### Step 9: Finalize the equation
Multiplying through by -1 gives the final equation:
\[ 8x + 4y - 18z - 29 = 0 \]

Thus, the equation of the set of points \( P \) such that the distances from points \( A \) and \( B \) are equal is:
\[ 8x + 4y - 18z - 29 = 0 \]