If `|z|= "max"{|z-2|,|z+2|}`, then

Options

A. `|z+barz|=2`
B. `z+barz=4`
C. `|z+barz|=1`
D. none of these

Explanation

To solve the problem \( |z| = \max\{|z-2|, |z+2|\} \), we will analyze the two cases separately.

### Step 1: Understand the condition
We need to find the values of \( z \) such that the modulus of \( z \) is equal to the maximum of the moduli \( |z-2| \) and \( |z+2| \).

### Step 2: Case 1: \( |z| = |z-2| \)
Assuming \( |z| = |z-2| \):
1. We square both sides:
\[ |z|^2 = |z-2|^2 \]
This gives:
\[ z \overline{z} = (z-2)(\overline{z}-2) \]
2. Expanding the right side:
\[ z \overline{z} = z \overline{z} - 2z - 2\overline{z} + 4 \]
3. Rearranging terms:
\[ 0 = -2z - 2\overline{z} + 4 \]
Simplifying gives:
\[ 2z + 2\overline{z} = 4 \quad \Rightarrow \quad z + \overline{z} = 2 \]
4. This implies:
\[ \text{Re}(z) = 1 \]
Thus, \( z \) can be expressed as:
\[ z = 1 + yi \quad \text{for any real } y \]

### Step 3: Case 2: \( |z| = |z+2| \)
Now assuming \( |z| = |z+2| \):
1. Again, we square both sides:
\[ |z|^2 = |z+2|^2 \]
This gives:
\[ z \overline{z} = (z+2)(\overline{z}+2) \]
2. Expanding the right side:
\[ z \overline{z} = z \overline{z} + 2z + 2\overline{z} + 4 \]
3. Rearranging terms:
\[ 0 = 2z + 2\overline{z} + 4 \]
Simplifying gives:
\[ 2z + 2\overline{z} = -4 \quad \Rightarrow \quad z + \overline{z} = -2 \]
4. This implies:
\[ \text{Re}(z) = -1 \]
Thus, \( z \) can be expressed as:
\[ z = -1 + yi \quad \text{for any real } y \]

### Step 4: Conclusion
From both cases, we find that:
- In Case 1, \( z = 1 + yi \)
- In Case 2, \( z = -1 + yi \)

Thus, the values of \( z \) satisfying the condition \( |z| = \max\{|z-2|, |z+2|\} \) are:
\[ z = 1 + yi \quad \text{or} \quad z = -1 + yi \quad \text{for any real } y. \]