If the argument of `(z-a)(barz-b)` is equal to that `((sqrt(3)+i)(1+sqrt(3)i)/(1+i))` where a,b,c are two real number and z is the complex conjugate o the complex number z find the locus of z in the rgand diagram. Find the value of a and b so that locus becomes a circle having its centre at `1/2(3+i)`

Options

A. `(3,2)`
B. `(2,1)`
C. `(2,3)`
D. `(2,4)`

Explanation

To solve the problem, we need to find the locus of the complex number \( z \) given that the argument of \( (z-a)(\overline{z}-b) \) is equal to the argument of the expression \( \frac{(\sqrt{3}+i)(1+\sqrt{3}i)}{1+i} \). We also need to find the values of \( a \) and \( b \) such that the locus is a circle centered at \( \frac{1}{2}(3+i) \).

### Step-by-Step Solution:

1. **Understanding the Expression**:
Given:
\[ \text{arg}((z-a)(\overline{z}-b)) = \text{arg}\left(\frac{(\sqrt{3}+i)(1+\sqrt{3}i)}{1+i}\right) \]

2. **Calculate the Right-Hand Side**:
First, we need to simplify the right-hand side:
- Calculate \( (\sqrt{3}+i)(1+\sqrt{3}i) \):
\[ = \sqrt{3} + 3i + i = \sqrt{3} + 3i + i = \sqrt{3} + 4i \]
- Now divide by \( 1+i \):
\[ \frac{\sqrt{3} + 4i}{1+i} \]
To divide, multiply the numerator and denominator by the conjugate of the denominator:
\[ = \frac{(\sqrt{3} + 4i)(1-i)}{(1+i)(1-i)} = \frac{\sqrt{3} - \sqrt{3}i + 4i + 4}{1^2 + 1^2} = \frac{(\sqrt{3} + 4) + (4 - \sqrt{3})i}{2} \]
- Thus, we have:
\[ = \frac{\sqrt{3} + 4}{2} + \frac{4 - \sqrt{3}}{2}i \]

3. **Finding the Argument**:
The argument of a complex number \( x + yi \) is given by:
\[ \text{arg}(x + yi) = \tan^{-1}\left(\frac{y}{x}\right) \]
Here, \( x = \frac{\sqrt{3} + 4}{2} \) and \( y = \frac{4 - \sqrt{3}}{2} \). Therefore:
\[ \text{arg}\left(\frac{\sqrt{3} + 4}{2} + \frac{4 - \sqrt{3}}{2}i\right) = \tan^{-1}\left(\frac{4 - \sqrt{3}}{\sqrt{3} + 4}\right) \]

4. **Setting Up the Locus**:
The argument condition gives us:
\[ \text{arg}((z-a)(\overline{z}-b)) = \text{arg}\left(\frac{\sqrt{3} + 4}{2} + \frac{4 - \sqrt{3}}{2}i\right) \]
This implies that the points \( z \) and \( \overline{z} \) are such that the line segments from \( a \) to \( z \) and from \( b \) to \( \overline{z} \) form a constant angle.

5. **Finding the Circle Center**:
The center of the circle is given as \( \frac{1}{2}(3+i) \). We can write:
\[ a + b = 3 \quad \text{(real part)} \]
\[ a - b = 1 \quad \text{(imaginary part)} \]

6. **Solving the System of Equations**:
From the equations:
- Adding:
\[ 2a = 4 \Rightarrow a = 2 \]
- Substituting \( a \) back:
\[ 2 + b = 3 \Rightarrow b = 1 \]

### Final Result:
Thus, the values of \( a \) and \( b \) are:
\[ \boxed{a = 2, b = 1} \]