A body moves in anticlockwise direction on a circular path in the x-y plane. The radius of the circular path is 5m and its centre is at the origin. In a certain interval of time, displacement of the body is obesrved to be 6m in the positive y-direction. Which of the following is true ?

Options

A. Its initial position vector is `5hati` m
B. Its initial position vector is `(-3hati+4hatj)` m.
C. Its final position vector is `(4hati+3hatj)`m.
D. Its final position vector is `6hatj` m.

Answer

To solve the problem, we need to analyze the motion of a body moving anticlockwise on a circular path centered at the origin (0,0) with a radius of 5 meters. The displacement of the body is given as 6 meters in the positive y-direction. We will determine the initial and final position vectors of the body based on the given information.

### Step-by-Step Solution:

1. **Understanding the Circular Path**:
- The body is moving anticlockwise on a circular path with a radius of 5 meters. This means the maximum distance from the center (origin) to any point on the path is 5 meters.

2. **Identifying the Displacement**:
- The displacement of the body is 6 meters in the positive y-direction. This indicates that the body has moved vertically upwards from its initial position.

3. **Setting Up the Coordinate System**:
- Let's define our coordinate system:
- The x-axis is horizontal, and the y-axis is vertical.
- The center of the circular path is at the origin (0,0).

4. **Using Pythagorean Theorem**:
- Since the radius of the circular path is 5 meters, we can use the Pythagorean theorem to find the position of the body.
- Let the initial position of the body be at point (x, y). The displacement of 6 meters in the positive y-direction means that if the body started at (x, y), its new position after the displacement will be (x, y + 6).

5. **Finding the Initial Position**:
- The initial position must satisfy the equation of the circle:
\[ x^2 + y^2 = 5^2 = 25 \]
- After the displacement, the new position (x, y + 6) must also satisfy the circle's equation:
\[ x^2 + (y + 6)^2 = 25 \]

6. **Setting Up the Equations**:
- From the first equation:
\[ x^2 + y^2 = 25 \quad (1) \]
- Expanding the second equation:
\[ x^2 + (y^2 + 12y + 36) = 25 \quad (2) \]
- From (2), we can substitute \(x^2\) from (1):
\[ 25 + 12y + 36 = 25 \]
- Simplifying gives:
\[ 12y + 36 = 0 \implies y = -3 \]

7. **Finding x**:
- Substitute \(y = -3\) back into equation (1):
\[ x^2 + (-3)^2 = 25 \implies x^2 + 9 = 25 \implies x^2 = 16 \implies x = 4 \text{ or } -4 \]
- Since the body is moving anticlockwise, we can choose \(x = 4\) (in the first quadrant).

8. **Initial Position Vector**:
- The initial position vector is:
\[ \text{Initial Position} = 4 \hat{i} - 3 \hat{j} \text{ meters} \]

9. **Final Position Vector**:
- After the displacement of 6 meters in the positive y-direction:
\[ \text{Final Position} = 4 \hat{i} + ( -3 + 6 ) \hat{j} = 4 \hat{i} + 3 \hat{j} \text{ meters} \]

10. **Conclusion**:
- The final position vector of the body is \(4 \hat{i} + 3 \hat{j}\) meters.

### Answer:
The correct option is: **Its final position vector is \(4 \hat{i} + 3 \hat{j}\) meters.**