A point P is moving in a plane such that the difference of its distances from two fixed points in the same plane is a constant. The path traced by the point P is a/an

Options

A. Circle
B. parabola
C. Ellipse
D. Hyperbola

Answer

To solve the problem, we need to understand the geometric definition of a hyperbola. The question states that a point \( P \) moves in a plane such that the difference of its distances from two fixed points (let's call them \( F_1 \) and \( F_2 \)) is a constant. We will analyze this step by step.

### Step-by-Step Solution:

1. **Identify the Fixed Points**:
Let \( F_1 \) and \( F_2 \) be the two fixed points in the plane.

2. **Define the Distances**:
Let \( d_1 \) be the distance from point \( P \) to point \( F_1 \) (i.e., \( PF_1 \)), and \( d_2 \) be the distance from point \( P \) to point \( F_2 \) (i.e., \( PF_2 \)).

3. **Set Up the Condition**:
According to the problem, the difference of these distances is constant:
\[ |d_2 - d_1| = k \]
where \( k \) is a constant.

4. **Understanding the Locus**:
The condition \( |d_2 - d_1| = k \) describes the locus of point \( P \). This is a well-known property of hyperbolas. In a hyperbola, the absolute difference of the distances from any point on the hyperbola to the two foci (which are analogous to our fixed points \( F_1 \) and \( F_2 \)) is constant.

5. **Conclusion**:
Therefore, the path traced by the point \( P \) is a hyperbola.

### Final Answer:
The path traced by point \( P \) is a **hyperbola**.