The equation `z bar(z) + ( 2-3i) z + ( 2+ 3i) bar(z) +4 =0` represents a circle of radius

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the equation \( z \bar{z} + (2 - 3i) z + (2 + 3i) \bar{z} + 4 = 0 \) and find the radius of the circle it represents, we can follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ z \bar{z} + (2 - 3i) z + (2 + 3i) \bar{z} + 4 = 0 \]
We can identify the terms in the equation that resemble the general form of a circle equation, which is:
\[ z \bar{z} + A z + A^* \bar{z} + B = 0 \]
where \( A = 2 - 3i \) and \( A^* = 2 + 3i \), and \( B = 4 \).

### Step 2: Identify the values of A, A*, and B
From the equation, we have:
- \( A = 2 - 3i \)
- \( A^* = 2 + 3i \)
- \( B = 4 \)

### Step 3: Calculate \( |A|^2 \)
To find the radius of the circle, we need to calculate \( |A|^2 \):
\[ |A|^2 = A A^* = (2 - 3i)(2 + 3i) \]
Using the formula for the product of conjugates:
\[ |A|^2 = 2^2 + (3)^2 = 4 + 9 = 13 \]

### Step 4: Calculate the radius
The radius \( r \) of the circle is given by the formula:
\[ r = \sqrt{|A|^2 - B} \]
Substituting the values we have:
\[ r = \sqrt{13 - 4} = \sqrt{9} = 3 \]

### Conclusion
Thus, the radius of the circle represented by the given equation is:
\[ \boxed{3} \]