Let `O=(0, 0), A=(3, 0), B=(0, -1)` and `C=(3, 2)`, then the minimum value of `|z|=|z-3|+|z+i|+|z-3-2i|` occurs at the (where, z is complex number)

Options

A. point of intersection of AB and CO
B. point of intersection of AC and BO
C. point of intersection of CB and AO
D. Mean of O, A, B, C

Answer

To find the minimum value of the expression \( |z| = |z - 3| + |z + i| + |z - (3 + 2i)| \), where \( z \) is a complex number, we can utilize the concept of triangle inequality in geometry.

### Step-by-Step Solution:

1. **Identify Points**:
We have the following points in the complex plane:
- \( O = (0, 0) \)
- \( A = (3, 0) \)
- \( B = (0, -1) \)
- \( C = (3, 2) \)

2. **Understanding the Expression**:
The expression \( |z| = |z - 3| + |z + i| + |z - (3 + 2i)| \) represents the sum of distances from the point \( z \) to the points \( A \), \( B \), and \( C \).

3. **Using Triangle Inequality**:
According to the triangle inequality, the minimum value of the sum of distances from point \( z \) to points \( A \), \( B \), and \( C \) occurs when \( z \) lies on the line segment connecting these points.

4. **Finding the Intersection**:
We need to find the intersection of the lines \( OA \) and \( BC \):
- The line \( OA \) connects points \( O(0, 0) \) and \( A(3, 0) \), which is the x-axis.
- The line \( BC \) connects points \( B(0, -1) \) and \( C(3, 2) \).

5. **Equation of Line BC**:
To find the equation of line \( BC \):
- The slope \( m \) of line \( BC \) is given by:
\[ m = \frac{y_C - y_B}{x_C - x_B} = \frac{2 - (-1)}{3 - 0} = \frac{3}{3} = 1 \]
- Using point-slope form, the equation of line \( BC \) can be written as:
\[ y - (-1) = 1(x - 0) \implies y = x - 1 \]

6. **Finding Intersection of Lines OA and BC**:
Since line \( OA \) is on the x-axis (where \( y = 0 \)), we set \( y = 0 \) in the equation of line \( BC \):
\[ 0 = x - 1 \implies x = 1 \]
Thus, the intersection point is \( (1, 0) \).

7. **Conclusion**:
Therefore, the minimum value of \( |z| \) occurs at the point \( z = 1 + 0i \) or simply \( z = 1 \).

### Final Answer:
The minimum value of \( |z| \) occurs at the point \( z = 1 \).