If `z_1 and z_2` are two complex numbers, then the equation of the perpendicular bisector of the segment `z_1 and z_2` is

Options

A. `( bar(z_2)- bar(z_1)) z + (z_2 - z_1) bar(z) = |z_2|^2 - |z_1|^2`
B. `( bar(z_2) - bar(z_1)) z = (z_2 - z_1 ) bar(z)`
C. `( bar(z_2) - bar(z_1)) z = |z_2|^2 - |z_1|^2`
D. None of these

Answer

To find the equation of the perpendicular bisector of the segment joining two complex numbers \( z_1 \) and \( z_2 \), we can follow these steps:

### Step 1: Understand the Geometric Meaning
The perpendicular bisector of a segment is the locus of points that are equidistant from the endpoints of the segment. In terms of complex numbers, this means that for any point \( z \) on the perpendicular bisector, the distances from \( z \) to \( z_1 \) and \( z_2 \) must be equal.

### Step 2: Set Up the Equation
We can express the condition of being equidistant mathematically:
\[ |z - z_1| = |z - z_2| \]
This states that the distance from \( z \) to \( z_1 \) is equal to the distance from \( z \) to \( z_2 \).

### Step 3: Square Both Sides
To eliminate the absolute values, we can square both sides of the equation:
\[ |z - z_1|^2 = |z - z_2|^2 \]

### Step 4: Expand Both Sides
Using the property that \( |a|^2 = a \cdot \overline{a} \), we expand both sides:
\[ (z - z_1)(\overline{z} - \overline{z_1}) = (z - z_2)(\overline{z} - \overline{z_2}) \]

### Step 5: Distribute
Expanding both sides gives:
\[ z\overline{z} - z\overline{z_1} - z_1\overline{z} + |z_1|^2 = z\overline{z} - z\overline{z_2} - z_2\overline{z} + |z_2|^2 \]

### Step 6: Simplify the Equation
Cancel \( z\overline{z} \) from both sides:
\[ -z\overline{z_1} - z_1\overline{z} + |z_1|^2 = -z\overline{z_2} - z_2\overline{z} + |z_2|^2 \]

Rearranging gives:
\[ z\overline{z_2} - z\overline{z_1} + z_1\overline{z} - z_2\overline{z} = |z_2|^2 - |z_1|^2 \]

### Step 7: Factor Out \( z \)
We can factor out \( z \) from the left side:
\[ z(\overline{z_2} - \overline{z_1}) + \overline{z}(z_1 - z_2) = |z_2|^2 - |z_1|^2 \]

### Step 8: Rearrange to Find the Equation
Rearranging gives us the equation of the perpendicular bisector:
\[ z(\overline{z_2} - \overline{z_1}) + \overline{z}(z_1 - z_2) = |z_2|^2 - |z_1|^2 \]

### Final Result
Thus, the equation of the perpendicular bisector of the segment joining the complex numbers \( z_1 \) and \( z_2 \) is:
\[ z(\overline{z_2} - \overline{z_1}) + \overline{z}(z_1 - z_2) = |z_2|^2 - |z_1|^2 \]