f A and B persons are moving with `V_(A) and V_(B)`. velocities in opposite directions. Magnitude of relative velocity of B w.r.t. A is x and magni-tude of relative velocity of A w.r.t B is y. Then

Options

A. `x gt y`
B. `x=y`
C. `x=2y`
D. `2x=y`

Answer

To solve the problem, we need to analyze the relative velocities of two persons A and B moving in opposite directions.

### Step-by-Step Solution:

1. **Understanding Relative Velocity**:
- The relative velocity of one object with respect to another is defined as the velocity of the first object minus the velocity of the second object.

2. **Define the Velocities**:
- Let the velocity of person A be \( V_A \) and the velocity of person B be \( V_B \).
- Since they are moving in opposite directions, we can assume \( V_A \) is positive and \( V_B \) is negative (or vice versa).

3. **Calculate Relative Velocity of B with respect to A**:
- The relative velocity of B with respect to A is given by:
\[ V_{BA} = V_B - V_A \]
- Since \( V_B \) is negative (let's say \( V_B = -|V_B| \)), we have:
\[ V_{BA} = -|V_B| - V_A = -(|V_B| + V_A) \]
- The magnitude of this relative velocity is:
\[ |V_{BA}| = |V_B| + |V_A| = x \]

4. **Calculate Relative Velocity of A with respect to B**:
- The relative velocity of A with respect to B is given by:
\[ V_{AB} = V_A - V_B \]
- Substituting \( V_B = -|V_B| \):
\[ V_{AB} = V_A - (-|V_B|) = V_A + |V_B| \]
- The magnitude of this relative velocity is:
\[ |V_{AB}| = |V_A| + |V_B| = y \]

5. **Relating the Magnitudes**:
- From the above calculations, we have:
\[ x = |V_B| + |V_A| \quad \text{(1)} \]
\[ y = |V_A| + |V_B| \quad \text{(2)} \]
- From equations (1) and (2), we can see that:
\[ x = y \]

6. **Conclusion**:
- Therefore, the relationship between the magnitudes of the relative velocities is:
\[ x = y \]

### Final Answer:
The correct relationship is \( x = y \).