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

Options

A. a) straight line
B. b) plane
C. c) sphere
D. d) None of these

Answer

To solve the problem, we need to find the locus of a point \( P(x, y) \) such that the difference of the squares of its distances from two fixed points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is constant.

### Step-by-Step Solution:

1. **Define the Points and Locus**:
Let the two fixed points be \( A(x_1, y_1) \) and \( B(x_2, y_2) \). We need to find the locus of point \( P(x, y) \).

2. **Distance Formulas**:
The distance from point \( P \) to point \( A \) is given by:
\[ d(P, A) = \sqrt{(x - x_1)^2 + (y - y_1)^2} \]
The distance from point \( P \) to point \( B \) is given by:
\[ d(P, B) = \sqrt{(x - x_2)^2 + (y - y_2)^2} \]

3. **Square of Distances**:
The squares of these distances are:
\[ d(P, A)^2 = (x - x_1)^2 + (y - y_1)^2 \]
\[ d(P, B)^2 = (x - x_2)^2 + (y - y_2)^2 \]

4. **Set Up the Equation**:
According to the problem, the difference of the squares of these distances is constant. We can express this as:
\[ d(P, A)^2 - d(P, B)^2 = k \]
where \( k \) is a constant.

5. **Substituting the Distance Squares**:
Substitute the expressions for the squares of the distances:
\[ \left((x - x_1)^2 + (y - y_1)^2\right) - \left((x - x_2)^2 + (y - y_2)^2\right) = k \]

6. **Expand the Equation**:
Expanding both sides, we get:
\[ (x^2 - 2xx_1 + x_1^2 + y^2 - 2yy_1 + y_1^2) - (x^2 - 2xx_2 + x_2^2 + y^2 - 2yy_2 + y_2^2) = k \]

7. **Simplifying the Equation**:
The \( x^2 \) and \( y^2 \) terms cancel out:
\[ -2xx_1 + x_1^2 - 2yy_1 + y_1^2 + 2xx_2 - x_2^2 + 2yy_2 - y_2^2 = k \]
Rearranging gives us:
\[ 2x(x_2 - x_1) + 2y(y_2 - y_1) + (x_1^2 - x_2^2 + y_1^2 - y_2^2) = k \]

8. **Final Form**:
This can be rearranged to show that it represents a linear equation in \( x \) and \( y \):
\[ 2x(x_2 - x_1) + 2y(y_2 - y_1) = k - (x_1^2 - x_2^2 + y_1^2 - y_2^2) \]
This is the equation of a plane in the coordinate system.

9. **Conclusion**:
Therefore, the locus of the point \( P \) is a plane.

### Final Answer:
The correct option is **B) plane**.