Consider the equaiton of line `abarz + baraz + b=0`, where b is a real parameter and a is fixed non-zero complex number.
The locus of mid-point of the line intercepted between real and imaginary axis is given by

Options

A. (a) `az- bar(az) =0`
B. (b) `az + bar(az) =0`
C. (c) `az-bar(az) + b =0`
D. (d) `az - bar(az) + 2b = 0`

Explanation

To solve the problem, we need to find the locus of the midpoint of the line segment intercepted between the real and imaginary axes given the equation of the line \( \overline{a} z + a z + b = 0 \), where \( b \) is a real parameter and \( a \) is a fixed non-zero complex number.

### Step-by-step Solution:

1. **Identify the Equation**:
The line is given by:
\[ \overline{a} z + a z + b = 0 \]

2. **Find the Intercept on the Real Axis**:
For the intercept on the real axis, we set \( z = x \) (where \( x \) is real). Thus, we have:
\[ \overline{a} x + a x + b = 0 \]
Rearranging gives:
\[ x(\overline{a} + a) = -b \]
Therefore, the intercept on the real axis \( z_R \) is:
\[ z_R = \frac{-b}{\overline{a} + a} \]

3. **Find the Intercept on the Imaginary Axis**:
For the intercept on the imaginary axis, we set \( z = iy \) (where \( y \) is real). Thus, we have:
\[ \overline{a} (iy) + a (iy) + b = 0 \]
Simplifying gives:
\[ i(\overline{a} - a)y + b = 0 \]
Rearranging gives:
\[ y = \frac{-b}{i(\overline{a} - a)} \]
Therefore, the intercept on the imaginary axis \( z_I \) is:
\[ z_I = \frac{-b}{i(\overline{a} - a)} \]

4. **Find the Midpoint of the Intercepted Segment**:
The midpoint \( z \) of the segment \( PQ \) (between \( z_R \) and \( z_I \)) is given by:
\[ z = \frac{z_R + z_I}{2} \]
Substituting the values of \( z_R \) and \( z_I \):
\[ z = \frac{\frac{-b}{\overline{a} + a} + \frac{-b}{i(\overline{a} - a)}}{2} \]

5. **Combine the Terms**:
Taking a common denominator:
\[ z = \frac{-b \left( \frac{1}{\overline{a} + a} + \frac{1}{i(\overline{a} - a)} \right)}{2} \]
Simplifying the expression inside the parentheses will yield a complex number.

6. **Express in Terms of \( b \)**:
After simplification, we can express \( z \) in terms of \( b \) and the fixed complex number \( a \).

7. **Determine the Locus**:
The locus of the midpoint as \( b \) varies will be a line or curve in the complex plane, which can be derived from the expression obtained in the previous step.

### Final Result:
The locus of the midpoint of the line intercepted between the real and imaginary axes is given by:
\[ \overline{a} z + a z + b = 0 \]
This can be further analyzed to find the specific geometric representation, which is typically a line in the complex plane.