An equation of straight line joining the complex numbers a and ib (where a, b `epsilon` R and a, b `ne` 0) is

Options

A. `z((1)/(a) - (i)/(b)) + bar(z) ((1)/(a) + (i)/(b)) = 2`
B. `z((1)/(a) - (i)/(b)) + bar(z) ((1)/(a) + (i)/(b)) = 2`
C. `z(a+ib)+bar(z)(a-ib)=2ab`
D. None of these

Answer

To find the equation of the straight line joining the complex numbers \( z_1 = a \) and \( z_2 = ib \) (where \( a, b \in \mathbb{R} \) and \( a, b \neq 0 \)), we can follow these steps:

### Step 1: Identify the Points
The complex numbers can be represented as points in the Cartesian plane:
- \( z_1 = a \) corresponds to the point \( (a, 0) \)
- \( z_2 = ib \) corresponds to the point \( (0, b) \)

### Step 2: Find the Slope of the Line
The slope \( m \) of the line joining the points \( (a, 0) \) and \( (0, b) \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{b - 0}{0 - a} = -\frac{b}{a} \]

### Step 3: Use the Point-Slope Form of the Line Equation
Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (a, 0) \) and \( m = -\frac{b}{a} \):
\[ y - 0 = -\frac{b}{a}(x - a) \]
This simplifies to:
\[ y = -\frac{b}{a}x + b \]

### Step 4: Convert to Complex Form
To express this in terms of complex numbers \( z \) and \( \overline{z} \):
Let \( z = x + iy \) and \( \overline{z} = x - iy \).
From the equation \( y = -\frac{b}{a}x + b \), we can replace \( x \) and \( y \):
- \( y = \frac{z - \overline{z}}{2i} \)
- \( x = \frac{z + \overline{z}}{2} \)

Substituting these into the line equation:
\[ \frac{z - \overline{z}}{2i} = -\frac{b}{a} \left( \frac{z + \overline{z}}{2} \right) + b \]

### Step 5: Simplify the Equation
Multiply through by \( 2i \) to eliminate the fraction:
\[ z - \overline{z} = -\frac{b}{a}(z + \overline{z}) \cdot i + 2bi \]
Rearranging gives:
\[ z(1 + \frac{bi}{a}) + \overline{z}(1 - \frac{bi}{a}) = 2bi \]

### Final Equation
The final equation of the line in complex form is:
\[ \frac{1}{a} z - \frac{i}{b} \overline{z} = 1 \]