Radius of the circle `z bar(z) + (2 + 5.5 i) z + (2-5.5 i) bar(z) + 4 = 0` is ___________

Answer

To find the radius of the circle given by the equation \( \overline{z} z + (2 + 5.5i) z + (2 - 5.5i) \overline{z} + 4 = 0 \), we can follow these steps:

### Step 1: Substitute \( z \) and \( \overline{z} \)
Let \( z = x + iy \) and \( \overline{z} = x - iy \).

### Step 2: Rewrite the equation
Substituting \( z \) and \( \overline{z} \) into the equation gives:
\[ (x - iy)(x + iy) + (2 + 5.5i)(x + iy) + (2 - 5.5i)(x - iy) + 4 = 0 \]

### Step 3: Expand the terms
1. The first term \( \overline{z} z = (x - iy)(x + iy) = x^2 + y^2 \).
2. The second term expands as:
\[ (2 + 5.5i)(x + iy) = 2x + 2iy + 5.5ix - 5.5y = (2x - 5.5y) + (2y + 5.5x)i \]
3. The third term expands as:
\[ (2 - 5.5i)(x - iy) = 2x - 2iy - 5.5ix - 5.5y = (2x - 5.5y) - (2y + 5.5x)i \]

### Step 4: Combine all terms
Combining all the terms:
\[ x^2 + y^2 + (2x - 5.5y) + (2y + 5.5x)i + (2x - 5.5y) - (2y + 5.5x)i + 4 = 0 \]
The imaginary parts cancel out, leading to:
\[ x^2 + y^2 + 4x - 11y + 4 = 0 \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ x^2 + 4x + y^2 - 11y + 4 = 0 \]

### Step 6: Completing the square
1. For \( x^2 + 4x \):
\[ x^2 + 4x = (x + 2)^2 - 4 \]
2. For \( y^2 - 11y \):
\[ y^2 - 11y = (y - \frac{11}{2})^2 - \frac{121}{4} \]

### Step 7: Substitute back into the equation
Substituting these back gives:
\[ (x + 2)^2 - 4 + (y - \frac{11}{2})^2 - \frac{121}{4} + 4 = 0 \]
This simplifies to:
\[ (x + 2)^2 + (y - \frac{11}{2})^2 - \frac{121}{4} = 0 \]
Rearranging gives:
\[ (x + 2)^2 + (y - \frac{11}{2})^2 = \frac{121}{4} \]

### Step 8: Identify the radius
The equation of the circle is in the form \( (x - a)^2 + (y - b)^2 = R^2 \), where \( R^2 = \frac{121}{4} \). Thus, the radius \( R \) is:
\[ R = \sqrt{\frac{121}{4}} = \frac{11}{2} = 5.5 \]

### Final Answer
The radius of the circle is \( 5.5 \).

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