The complex number z satisfying `| z - 1| = | z - 3| = | z - i | ` is

Options

A. 2 +i
B. `(3)/(2) + (1)/(2)i`
C. `2 + 2i`
D. none of these

Answer

To solve the problem of finding the complex number \( z \) that satisfies the conditions \( |z - 1| = |z - 3| = |z - i| \), we can follow these steps:

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

### Step 2: Set up the equations
The given conditions can be expressed as:
1. \( |z - 1| = |z - 3| \)
2. \( |z - 1| = |z - i| \)

### Step 3: Solve the first equation
From the first condition:
\[ |z - 1| = |z - 3| \]
This translates to:
\[ | (x + iy) - 1 | = | (x + iy) - 3 | \]
Calculating the moduli, we have:
\[ \sqrt{(x - 1)^2 + y^2} = \sqrt{(x - 3)^2 + y^2} \]
Squaring both sides:
\[ (x - 1)^2 + y^2 = (x - 3)^2 + y^2 \]
The \( y^2 \) terms cancel out:
\[ (x - 1)^2 = (x - 3)^2 \]
Expanding both sides:
\[ x^2 - 2x + 1 = x^2 - 6x + 9 \]
Simplifying gives:
\[ -2x + 1 = -6x + 9 \]
Rearranging:
\[ 4x = 8 \implies x = 2 \]

### Step 4: Substitute \( x \) back into the second equation
Now we substitute \( x = 2 \) into the second condition:
\[ |z - 1| = |z - i| \]
This becomes:
\[ | (2 + iy) - 1 | = | (2 + iy) - i | \]
Calculating the moduli:
\[ | (1 + iy) | = | (2 - i + iy) | \]
This translates to:
\[ \sqrt{1^2 + y^2} = \sqrt{2^2 + (y - 1)^2} \]
Squaring both sides:
\[ 1 + y^2 = 4 + (y - 1)^2 \]
Expanding the right side:
\[ 1 + y^2 = 4 + (y^2 - 2y + 1) \]
Simplifying gives:
\[ 1 + y^2 = 4 + y^2 - 2y + 1 \]
Cancelling \( y^2 \) and simplifying:
\[ 1 = 5 - 2y \]
Rearranging gives:
\[ 2y = 4 \implies y = 2 \]

### Step 5: Write the final solution
Thus, we have \( x = 2 \) and \( y = 2 \). Therefore, the complex number \( z \) is:
\[ z = 2 + 2i \]

### Final Answer
The complex number \( z \) satisfying the given conditions is:
\[ \boxed{2 + 2i} \]