Consider the equation `az + b bar(z) + c =0`, where a,b,c `in`Z
If `|a| ne |b|`, then z represents

Options

A. circle
B. straight line
C. one point
D. ellispe

Explanation

To solve the equation \( az + b \overline{z} + c = 0 \) where \( a, b, c \in \mathbb{Z} \) and \( |a| \neq |b| \), we will follow these steps:

### Step 1: Write down the original equation
We start with the equation:
\[ az + b \overline{z} + c = 0 \]

### Step 2: Take the conjugate of the equation
Taking the conjugate of the entire equation gives us:
\[ \overline{az + b \overline{z} + c} = 0 \]
Using the properties of conjugates, we can separate this into:
\[ \overline{az} + \overline{b \overline{z}} + \overline{c} = 0 \]
This simplifies to:
\[ \overline{a} \overline{z} + \overline{b} z + \overline{c} = 0 \]

### Step 3: Substitute \( \overline{z} \) from the original equation
From the original equation \( az + b \overline{z} + c = 0 \), we can express \( \overline{z} \) as:
\[ \overline{z} = \frac{-c - az}{b} \]
Substituting this into the conjugate equation gives:
\[ \overline{a} \overline{z} + \overline{b} z + \overline{c} = 0 \]
Substituting for \( \overline{z} \):
\[ \overline{a} \left( \frac{-c - az}{b} \right) + \overline{b} z + \overline{c} = 0 \]

### Step 4: Multiply through by \( b \) to eliminate the fraction
Multiplying the entire equation by \( b \) gives:
\[ -\overline{a}c - a z + \overline{b} b z + \overline{c} b = 0 \]

### Step 5: Rearranging the equation
Rearranging the terms leads to:
\[ (-\overline{a}c + \overline{c}b) + (|\overline{b}|^2 - |a|^2)z = 0 \]

### Step 6: Solve for \( z \)
We can isolate \( z \):
\[ z = \frac{\overline{a}c - \overline{c}b}{|b|^2 - |a|^2} \]

### Step 7: Analyze the condition \( |a| \neq |b| \)
Given that \( |a| \neq |b| \), the denominator \( |b|^2 - |a|^2 \) is non-zero. Therefore, \( z \) is uniquely defined.

### Conclusion
Since \( z \) is expressed as a single point in the complex plane, we conclude that \( z \) represents one point in the Argand plane.

### Final Answer
Thus, the answer is that \( z \) represents one point in the Argand plane.

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