Slope of line `zbar(a)+barzoverset(.)(a)+b=0, b in R ` is -

Options

A. `(-Re(a))/(Im(a))`
B. `(-Im(a))/(Re(a))`
C. `(Re(a))/(Im(a))`
D. None of these

Explanation

To find the slope of the line given by the equation \( \overline{z}(a) + \overline{z}(a) + b = 0 \), where \( b \in \mathbb{R} \), we can follow these steps:

### Step 1: Express \( z \) in terms of real and imaginary parts
Let \( z = x + i y \), where \( x \) is the real part and \( y \) is the imaginary part of the complex number \( z \).

### Step 2: Substitute \( z \) into the equation
Substituting \( z \) into the equation gives us:
\[ \overline{z} = x - i y \]
Thus, the equation becomes:
\[ (x - i y)(a) + (x + i y)(a) + b = 0 \]

### Step 3: Simplify the equation
Expanding the equation, we have:
\[ (xa - i ya) + (xa + i ya) + b = 0 \]
This simplifies to:
\[ 2xa + b = 0 \]
The imaginary parts cancel out.

### Step 4: Rearranging to find the slope
Now, we can rearrange the equation to isolate \( y \):
\[ 2xa = -b \]
This implies:
\[ y = -\frac{b}{2a} \]

### Step 5: Identify the slope
In the context of a line in the Cartesian plane, the slope \( m \) can be identified from the equation \( y = mx + c \). Here, the slope \( m \) is given by:
\[ m = -\frac{b}{2a} \]

### Conclusion
Thus, the slope of the line is:
\[ -\frac{b}{2a} \]

---