The equation `z bar(z) + a bar(z) + bar(a) z + b = 0, b in R ` represents a circle if

Options

A. `|a|^(2) = b`
B. ` |a^(2)| gt b`
C. ` |a|^(2) lt b`
D. none of these

Answer

To determine the conditions under which the equation \( z \bar{z} + \bar{a} z + a \bar{z} + b = 0 \) represents a circle, we can follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation:
\[ z \bar{z} + \bar{a} z + a \bar{z} + b = 0 \]
Here, \( z \) is a complex number, and \( \bar{z} \) is its conjugate.

### Step 2: Identify the Form of the Circle
The general form of a circle in the complex plane can be expressed as:
\[ |z - z_0|^2 = r^2 \]
where \( z_0 \) is the center of the circle and \( r \) is the radius.

### Step 3: Rearranging the Terms
We can rearrange the given equation:
\[ z \bar{z} + \bar{a} z + a \bar{z} = -b \]
This can be rewritten as:
\[ z \bar{z} + (\bar{a} z + a \bar{z}) = -b \]

### Step 4: Express in Terms of Modulus
To express this in a form that resembles the circle equation, we can use the identity \( z \bar{z} = |z|^2 \):
\[ |z|^2 + (\bar{a} z + a \bar{z}) + b = 0 \]

### Step 5: Identify the Center and Radius
From the standard form of the circle, we can identify:
- The center \( z_0 = -\frac{\bar{a}}{2} \)
- The radius \( r^2 = \left(\frac{|\bar{a}|^2}{4} - b\right) \)

### Step 6: Condition for a Circle
For the equation to represent a circle, the radius must be positive:
\[ \frac{|\bar{a}|^2}{4} - b > 0 \]
This simplifies to:
\[ |\bar{a}|^2 > 4b \]
Since \( |\bar{a}|^2 = |a|^2 \), we can rewrite this as:
\[ |a|^2 > 4b \]

### Final Result
Thus, the condition for the equation \( z \bar{z} + \bar{a} z + a \bar{z} + b = 0 \) to represent a circle is:
\[ |a|^2 > 4b \]

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