Consider the equaiton of line `abarz + abarz+ abarz + b=0`, where b is a real parameter and a is fixed non-zero complex number.
The intercept of line on real axis is given by

Options

A. `(-2b)/(a+bara)`
B. `(-b)/(2(a+ bara))`
C. `(-b)/(a+bara)`
D. `(b)/(a+bara)`

Explanation

To find the intercept of the line given by the equation \( a \bar{z} + \bar{a} z + b = 0 \) on the real axis, we will follow these steps:

### Step 1: Understand the components of the equation
The equation involves complex numbers where \( z \) is a complex variable and \( \bar{z} \) is its conjugate. The parameter \( b \) is a real number, and \( a \) is a fixed non-zero complex number.

### Step 2: Substitute \( z \) with its real part
To find the intercept on the real axis, we can set \( z = x \) where \( x \) is a real number. Thus, \( \bar{z} = x \) as well. The equation simplifies to:
\[ a \bar{z} + \bar{a} z + b = 0 \implies a x + \bar{a} x + b = 0 \]

### Step 3: Factor out \( x \)
We can factor out \( x \) from the equation:
\[ x(a + \bar{a}) + b = 0 \]

### Step 4: Solve for \( x \)
Rearranging the equation gives:
\[ x(a + \bar{a}) = -b \]
Thus,
\[ x = -\frac{b}{a + \bar{a}} \]

### Step 5: Identify the intercept
The value of \( x \) represents the intercept of the line on the real axis. Therefore, the intercept is:
\[ \text{Intercept} = -\frac{b}{a + \bar{a}} \]

### Final Answer
The intercept of the line on the real axis is given by:
\[ -\frac{b}{a + \bar{a}} \]

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