Let `3-i` and `2+i` be affixes of two points A and B in the Argand plane and P represents the complex number `z=x+iy`. Then, the locus of the P if `|z-3+i|=|z-2-i|`, is

Options

A. circle on AB as diameter
B. the line AB
C. the perpendicular bisector of AB
D. none of these

Answer

To find the locus of the point \( P \) represented by the complex number \( z = x + iy \) such that \( |z - (3 - i)| = |z - (2 + i)| \), we can follow these steps:

### Step 1: Identify the points A and B
The points \( A \) and \( B \) are given as:
- \( A = 3 - i \) which corresponds to the coordinates \( (3, -1) \)
- \( B = 2 + i \) which corresponds to the coordinates \( (2, 1) \)

### Step 2: Rewrite the equation
The equation \( |z - (3 - i)| = |z - (2 + i)| \) can be rewritten as:
\[ |z - A| = |z - B| \]

### Step 3: Interpret the equation geometrically
The equation \( |z - A| = |z - B| \) means that the point \( P \) is equidistant from points \( A \) and \( B \). This describes the locus of points that are equidistant from two fixed points, which is the perpendicular bisector of the line segment \( AB \).

### Step 4: Find the midpoint of segment AB
The midpoint \( M \) of segment \( AB \) can be calculated as:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{3 + 2}{2}, \frac{-1 + 1}{2} \right) = \left( \frac{5}{2}, 0 \right) \]

### Step 5: Determine the slope of line AB
The slope \( m \) of line \( AB \) can be calculated as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-1)}{2 - 3} = \frac{2}{-1} = -2 \]

### Step 6: Find the slope of the perpendicular bisector
The slope of the perpendicular bisector will be the negative reciprocal of the slope of \( AB \):
\[ m_{perpendicular} = \frac{1}{2} \]

### Step 7: Write the equation of the perpendicular bisector
Using the point-slope form of the line equation, the equation of the perpendicular bisector can be written as:
\[ y - y_0 = m_{perpendicular}(x - x_0) \]
Substituting the midpoint \( M\left(\frac{5}{2}, 0\right) \) into the equation:
\[ y - 0 = \frac{1}{2}\left(x - \frac{5}{2}\right) \]
Simplifying this gives:
\[ y = \frac{1}{2}x - \frac{5}{4} \]

### Conclusion
The locus of the point \( P \) is the line represented by the equation:
\[ y = \frac{1}{2}x - \frac{5}{4} \]
This line is the perpendicular bisector of the segment joining points \( A \) and \( B \).

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