A moving point 'P' moves such that `AP^(2)+BP^(2)=10` where the co-ordinates of the points A and B are respectively (2,3,-4) and (0,0,1).

Explanation

To find the locus of the moving point \( P \) such that \( AP^2 + BP^2 = 10 \), where the coordinates of points \( A \) and \( B \) are \( A(2, 3, -4) \) and \( B(0, 0, 1) \), we will follow these steps:

### Step 1: Write the distance formulas for \( AP \) and \( BP \)

Using the distance formula, we can express \( AP \) and \( BP \) as follows:

- The distance \( AP \) from point \( A(2, 3, -4) \) to point \( P(x, y, z) \) is given by:
\[ AP^2 = (x - 2)^2 + (y - 3)^2 + (z + 4)^2 \]

- The distance \( BP \) from point \( B(0, 0, 1) \) to point \( P(x, y, z) \) is given by:
\[ BP^2 = (x - 0)^2 + (y - 0)^2 + (z - 1)^2 = x^2 + y^2 + (z - 1)^2 \]

### Step 2: Set up the equation based on the given condition

According to the problem, we have:
\[ AP^2 + BP^2 = 10 \]
Substituting the expressions for \( AP^2 \) and \( BP^2 \) into this equation gives:
\[ (x - 2)^2 + (y - 3)^2 + (z + 4)^2 + x^2 + y^2 + (z - 1)^2 = 10 \]

### Step 3: Expand the equation

Now, we will expand each term:
1. \( (x - 2)^2 = x^2 - 4x + 4 \)
2. \( (y - 3)^2 = y^2 - 6y + 9 \)
3. \( (z + 4)^2 = z^2 + 8z + 16 \)
4. \( (z - 1)^2 = z^2 - 2z + 1 \)

Substituting these expansions back into the equation gives:
\[ (x^2 - 4x + 4) + (y^2 - 6y + 9) + (z^2 + 8z + 16) + x^2 + y^2 + (z^2 - 2z + 1) = 10 \]

### Step 4: Combine like terms

Combining like terms:
- For \( x^2 \): \( 2x^2 \)
- For \( y^2 \): \( 2y^2 \)
- For \( z^2 \): \( 2z^2 \)
- For \( x \): \( -4x \)
- For \( y \): \( -6y \)
- For \( z \): \( 8z - 2z = 6z \)
- Constant terms: \( 4 + 9 + 16 + 1 = 30 \)

So we have:
\[ 2x^2 + 2y^2 + 2z^2 - 4x - 6y + 6z + 30 = 10 \]

### Step 5: Simplify the equation

Now, we simplify this equation:
\[ 2x^2 + 2y^2 + 2z^2 - 4x - 6y + 6z + 20 = 0 \]
Dividing the entire equation by 2 gives:
\[ x^2 + y^2 + z^2 - 2x - 3y + 3z + 10 = 0 \]

### Step 6: Final equation of the locus

This is the equation of the locus of point \( P \):
\[ x^2 + y^2 + z^2 - 2x - 3y + 3z + 10 = 0 \]