A straight line is passing through thte points represented by the complex number a+ ib and `(1)/(-1+ib)` , where `(a,b) ne (0,0).`
Which one of the following is correct ?

Options

A. It passes thro ugh the origin
B. It is parallel to the x-axis
C. It is parallel to the y-axis
D. It parallel to the y-axis

Explanation

To solve the problem step by step, we need to analyze the points represented by the complex numbers and derive the equation of the line passing through these points.

### Step 1: Identify the Points
The points represented by the complex numbers are:
1. \( z_1 = a + ib \)
2. \( z_2 = \frac{1}{-1 + ib} \)

### Step 2: Rationalize the Second Point
To find \( z_2 \), we rationalize the denominator:
\[ z_2 = \frac{1}{-1 + ib} \cdot \frac{-1 - ib}{-1 - ib} = \frac{-1 - ib}{(-1)^2 + (b)^2} = \frac{-1 - ib}{1 + b^2} \]
Thus, we can express \( z_2 \) as:
\[ z_2 = \frac{-1}{1 + b^2} - i \frac{b}{1 + b^2} \]

### Step 3: Extract Real and Imaginary Parts
From the two points, we have:
- For \( z_1 = a + ib \), the real part is \( a \) and the imaginary part is \( b \).
- For \( z_2 = \frac{-1}{1 + b^2} - i \frac{b}{1 + b^2} \), the real part is \( \frac{-1}{1 + b^2} \) and the imaginary part is \( \frac{-b}{1 + b^2} \).

### Step 4: Write the Equation of the Line
Using the two points \( (a, b) \) and \( \left(\frac{-1}{1 + b^2}, \frac{-b}{1 + b^2}\right) \), we can find the slope \( m \) of the line:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\frac{-b}{1 + b^2} - b}{\frac{-1}{1 + b^2} - a} \]

### Step 5: Simplify the Equation
Substituting the values:
\[ y - b = m(x - a) \]
This gives us the equation of the line in point-slope form.

### Step 6: Check if the Line Passes Through the Origin
To check if the line passes through the origin (0, 0), substitute \( x = 0 \) and \( y = 0 \) into the line equation:
\[ 0 - b = m(0 - a) \]
If this holds true, then the line passes through the origin.

### Step 7: Determine the Correct Option
After analyzing the equation, we can conclude which option is correct based on whether the line passes through the origin or is parallel to the axes.