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 \( |z - 2| = \min \{ |z - 1|, |z - 3| \} \), where \( z \) is a complex number, we will analyze the two cases where the minimum can occur.

### Step 1: Define the complex number
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Write the expressions for the moduli
We can express the moduli as follows:
- \( |z - 2| = |(x + iy) - 2| = |(x - 2) + iy| = \sqrt{(x - 2)^2 + y^2} \)
- \( |z - 1| = |(x + iy) - 1| = |(x - 1) + iy| = \sqrt{(x - 1)^2 + y^2} \)
- \( |z - 3| = |(x + iy) - 3| = |(x - 3) + iy| = \sqrt{(x - 3)^2 + y^2} \)

### Step 3: Case 1 - When \( |z - 1| \) is the minimum
Assume \( |z - 2| = |z - 1| \):
\[ \sqrt{(x - 2)^2 + y^2} = \sqrt{(x - 1)^2 + y^2} \]
Squaring both sides:
\[ (x - 2)^2 + y^2 = (x - 1)^2 + y^2 \]
Cancelling \( y^2 \):
\[ (x - 2)^2 = (x - 1)^2 \]
Expanding both sides:
\[ x^2 - 4x + 4 = x^2 - 2x + 1 \]
Simplifying:
\[ -4x + 4 = -2x + 1 \implies -2x = -3 \implies x = \frac{3}{2} \]

### Step 4: Case 2 - When \( |z - 3| \) is the minimum
Assume \( |z - 2| = |z - 3| \):
\[ \sqrt{(x - 2)^2 + y^2} = \sqrt{(x - 3)^2 + y^2} \]
Squaring both sides:
\[ (x - 2)^2 + y^2 = (x - 3)^2 + y^2 \]
Cancelling \( y^2 \):
\[ (x - 2)^2 = (x - 3)^2 \]
Expanding both sides:
\[ x^2 - 4x + 4 = x^2 - 6x + 9 \]
Simplifying:
\[ -4x + 4 = -6x + 9 \implies 2x = 5 \implies x = \frac{5}{2} \]

### Conclusion
The real part of \( z \) can be either \( \frac{3}{2} \) or \( \frac{5}{2} \). Thus, the solution to the problem is:
\[ \text{The real part of } z \text{ can be } \frac{3}{2} \text{ or } \frac{5}{2}. \]