Find the radius and centre of the circle `z bar(z) + (1-i) z + (1+ i) bar(z)- 7 = 0`

Answer

To find the radius and center of the circle given by the equation \( \overline{z} z + (1-i) z + (1+i) \overline{z} - 7 = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ \overline{z} z + (1-i) z + (1+i) \overline{z} - 7 = 0 \]
We can rearrange it to match the general form of a circle.

### Step 2: Identify the general form of the circle
The general equation of a circle in the complex plane is:
\[ z \overline{z} + a \overline{z} + \overline{a} z + b = 0 \]
where \( a \) is a complex number related to the coefficients of \( z \) and \( \overline{z} \), and \( b \) is a constant.

### Step 3: Compare coefficients
From the equation:
- \( a = 1 - i \)
- \( \overline{a} = 1 + i \)
- \( b = -7 \)

### Step 4: Calculate the center
The center \( C \) of the circle can be found using the formula:
\[ C = -\frac{a}{2} \]
Substituting \( a = 1 - i \):
\[ C = -\frac{1 - i}{2} = -\frac{1}{2} + \frac{i}{2} \]
Thus, the center is:
\[ C = -\frac{1}{2} + \frac{i}{2} \]

### Step 5: Calculate the radius
The radius \( r \) can be calculated using the formula:
\[ r = \sqrt{|a|^2 - b} \]
First, we find \( |a|^2 \):
\[ |a|^2 = |1 - i|^2 = 1^2 + (-1)^2 = 1 + 1 = 2 \]
Now substituting into the radius formula:
\[ r = \sqrt{2 - (-7)} = \sqrt{2 + 7} = \sqrt{9} = 3 \]

### Final Result
The center of the circle is:
\[ C = -\frac{1}{2} + \frac{i}{2} \]
The radius of the circle is:
\[ r = 3 \]