If `|z-2|= "min" {|z-1|,|z-3|}`, where z is a complex number, then

Options

A. Re `(z) = 3//2`
B. Re (z) ` = 5//2`
C. Re `(z) in {(3)/(2), (5)/(2)}`
D. None of these

Answer

To solve the problem, we need to analyze the condition given:

\[ |z - 2| = \min\{|z - 1|, |z - 3|\} \]

where \( z \) is a complex number.

### Step-by-step Solution:

1. **Understanding the Condition**:
The equation states that the distance from the complex number \( z \) to the point \( 2 \) is equal to the minimum of the distances from \( z \) to the points \( 1 \) and \( 3 \).

2. **Assuming Cases**:
We will consider two cases based on the definition of minimum:
- Case 1: \( |z - 2| = |z - 1| \)
- Case 2: \( |z - 2| = |z - 3| \)

3. **Case 1: \( |z - 2| = |z - 1| \)**:
- Let \( z = x + iy \) where \( x \) and \( y \) are real numbers.
- Then, the equation becomes:
\[ |(x - 2) + iy| = |(x - 1) + iy| \]
- Squaring both sides gives:
\[ (x - 2)^2 + y^2 = (x - 1)^2 + y^2 \]
- Simplifying this:
\[ (x - 2)^2 = (x - 1)^2 \]
\[ x^2 - 4x + 4 = x^2 - 2x + 1 \]
\[ -4x + 4 = -2x + 1 \]
\[ -2x = -3 \implies x = \frac{3}{2} \]

4. **Case 2: \( |z - 2| = |z - 3| \)**:
- Similarly, we consider:
\[ |(x - 2) + iy| = |(x - 3) + iy| \]
- Squaring both sides gives:
\[ (x - 2)^2 + y^2 = (x - 3)^2 + y^2 \]
- Simplifying this:
\[ (x - 2)^2 = (x - 3)^2 \]
\[ x^2 - 4x + 4 = x^2 - 6x + 9 \]
\[ -4x + 4 = -6x + 9 \]
\[ 2x = 5 \implies x = \frac{5}{2} \]

5. **Finding the Minimum**:
- From Case 1, we found \( x = \frac{3}{2} \).
- From Case 2, we found \( x = \frac{5}{2} \).
- Since we need the minimum value, we take \( x = \frac{3}{2} \).

### Final Answer:
The real part of \( z \) is \( \frac{3}{2} \).