The locus of a point which moves so that the difference of the squares of its distances from two given points is constant, is a

Options

A. straight line
B. plane
C. sphere
D. none of these

Explanation

To find the locus of a point that moves such that the difference of the squares of its distances from two given points is constant, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Given Points and the Moving Point**:
Let the two fixed points be \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \). Let the moving point be \( P(x, y, z) \).

2. **Write the Distance Formulas**:
The distance from point \( P \) to point \( A \) is given by:
\[ d_A = \sqrt{(x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2} \]
The distance from point \( P \) to point \( B \) is given by:
\[ d_B = \sqrt{(x - x_2)^2 + (y - y_2)^2 + (z - z_2)^2} \]

3. **Square the Distances**:
The squares of these distances are:
\[ d_A^2 = (x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2 \]
\[ d_B^2 = (x - x_2)^2 + (y - y_2)^2 + (z - z_2)^2 \]

4. **Set Up the Equation**:
According to the problem, the difference of the squares of the distances is constant:
\[ d_A^2 - d_B^2 = k \]
where \( k \) is a constant.

5. **Substitute the Squared Distances**:
Substitute the expressions for \( d_A^2 \) and \( d_B^2 \):
\[ \left((x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2\right) - \left((x - x_2)^2 + (y - y_2)^2 + (z - z_2)^2\right) = k \]

6. **Expand the Squares**:
Expanding both sides gives:
\[ (x^2 - 2xx_1 + x_1^2 + y^2 - 2yy_1 + y_1^2 + z^2 - 2zz_1 + z_1^2) - (x^2 - 2xx_2 + x_2^2 + y^2 - 2yy_2 + y_2^2 + z^2 - 2zz_2 + z_2^2) = k \]

7. **Simplify the Equation**:
Cancel out the \( x^2, y^2, z^2 \) terms:
\[ -2xx_1 + x_1^2 - (-2xx_2 + x_2^2) - 2yy_1 + y_1^2 - (-2yy_2 + y_2^2) - 2zz_1 + z_1^2 - (-2zz_2 + z_2^2) = k \]
This simplifies to:
\[ 2x(x_2 - x_1) + 2y(y_2 - y_1) + 2z(z_2 - z_1) = k + (x_1^2 - x_2^2 + y_1^2 - y_2^2 + z_1^2 - z_2^2) \]

8. **Rearranging the Equation**:
Rearranging gives:
\[ 2x(x_2 - x_1) + 2y(y_2 - y_1) + 2z(z_2 - z_1) = C \]
where \( C = k + (x_1^2 - x_2^2 + y_1^2 - y_2^2 + z_1^2 - z_2^2) \).

9. **Final Form**:
Dividing through by 2 gives:
\[ x(x_2 - x_1) + y(y_2 - y_1) + z(z_2 - z_1) = \frac{C}{2} \]
This represents a plane in 3D space.

### Conclusion:
The locus of the point \( P(x, y, z) \) is a plane.