Consider the region `R` in the Argand plane described by the complex number. `Z` satisfying the inequalities `|Z-2| le |Z-4|`, `|Z-3| le |Z+3|`, `|Z-i| le |Z-3i|`, `|Z+i| le |Z+3i|`
Answer the followin questions :
Minimum of `|Z_(1)-Z_(2)|` given that `Z_(1)`, `Z_(2)` are any two complex numbers lying in the region `R` is

Options

A. `0`
B. `5`
C. `sqrt(13)`
D. `3`

Explanation

To solve the problem, we need to analyze the inequalities that define the region \( R \) in the Argand plane and then find the minimum of \( |Z_1 - Z_2| \) for any two complex numbers \( Z_1 \) and \( Z_2 \) that lie within this region.

### Step-by-Step Solution:

1. **Understanding the inequalities**:
- The inequalities given are:
1. \( |Z - 2| \leq |Z - 4| \)
2. \( |Z - 3| \leq |Z + 3| \)
3. \( |Z - i| \leq |Z - 3i| \)
4. \( |Z + i| \leq |Z + 3i| \)

2. **Interpreting the first inequality**:
- The inequality \( |Z - 2| \leq |Z - 4| \) means that the point \( Z \) is closer to \( 2 \) than to \( 4 \). This describes the region on the Argand plane that is on or to the left of the perpendicular bisector of the line segment joining \( 2 \) and \( 4 \).

3. **Interpreting the second inequality**:
- The inequality \( |Z - 3| \leq |Z + 3| \) indicates that \( Z \) is closer to \( 3 \) than to \( -3 \). This describes the region on the Argand plane that is on or to the right of the perpendicular bisector of the line segment joining \( 3 \) and \( -3 \).

4. **Interpreting the third inequality**:
- The inequality \( |Z - i| \leq |Z - 3i| \) means that \( Z \) is closer to \( i \) than to \( 3i \). This describes the region on the Argand plane that is on or below the perpendicular bisector of the line segment joining \( i \) and \( 3i \).

5. **Interpreting the fourth inequality**:
- The inequality \( |Z + i| \leq |Z + 3i| \) indicates that \( Z \) is closer to \( -i \) than to \( -3i \). This describes the region on the Argand plane that is on or above the perpendicular bisector of the line segment joining \( -i \) and \( -3i \).

6. **Finding the intersection of the regions**:
- The region \( R \) is the intersection of all the regions described by the inequalities. This region is bounded and can be visualized as a polygon in the Argand plane.

7. **Finding the minimum of \( |Z_1 - Z_2| \)**:
- Since \( Z_1 \) and \( Z_2 \) are both in the region \( R \), the minimum distance between any two points in a bounded region occurs when the two points coincide. Therefore, the minimum value of \( |Z_1 - Z_2| \) is \( 0 \).

### Conclusion:
The minimum of \( |Z_1 - Z_2| \) given that \( Z_1 \) and \( Z_2 \) are any two complex numbers lying in the region \( R \) is \( 0 \).