For a complex number Z, if the argument of `(Z-a)(barZ-b)` is `(pi)/(4)` or `(-3pi)/(4)` (where a, b are two real numbers), then the value of ab such that the locus of Z represents a circle with centre `(3)/(2)+(i)/(2)` is (where, `i^(2)=-1`)

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to analyze the given conditions for the complex number \( Z \) and derive the necessary equations step by step.

### Step 1: Understand the Given Information
We are given that the argument of \( (Z - a)(\overline{Z} - b) \) is either \( \frac{\pi}{4} \) or \( -\frac{3\pi}{4} \). This means that the ratio of the imaginary part to the real part of \( (Z - a)(\overline{Z} - b) \) must equal \( 1 \) or \( -1 \).

### Step 2: Express \( Z \) in Terms of Its Components
Let \( Z = x + iy \), where \( x \) and \( y \) are real numbers. Then, we have:
\[ \overline{Z} = x - iy \]
Thus, we can rewrite the expression:
\[ (Z - a)(\overline{Z} - b) = (x + iy - a)(x - iy - b) \]

### Step 3: Expand the Expression
Expanding the product:
\[ = ((x - a) + iy)((x - b) - iy) = (x - a)(x - b) + (x - a)(-iy) + iy(x - b) + y^2 \]
This simplifies to:
\[ = (x - a)(x - b) + y^2 + i(y(x - b) - (x - a)y) \]
\[ = (x - a)(x - b) + y^2 + i(y(x - b - (x - a))) \]
\[ = (x - a)(x - b) + y^2 + i(y(a - b)) \]

### Step 4: Identify Real and Imaginary Parts
The real part is:
\[ (x - a)(x - b) + y^2 \]
The imaginary part is:
\[ y(a - b) \]

### Step 5: Set Up the Argument Conditions
Since the argument is \( \frac{\pi}{4} \) or \( -\frac{3\pi}{4} \), we have:
1. For \( \frac{\pi}{4} \):
\[ \frac{y(a - b)}{(x - a)(x - b) + y^2} = 1 \implies y(a - b) = (x - a)(x - b) + y^2 \]
2. For \( -\frac{3\pi}{4} \):
\[ \frac{y(a - b)}{(x - a)(x - b) + y^2} = -1 \implies y(a - b) = -((x - a)(x - b) + y^2) \]

### Step 6: Solve for \( a \) and \( b \)
From the two conditions, we can derive:
1. \( y(a - b) = (x - a)(x - b) + y^2 \)
2. \( y(a - b) = -((x - a)(x - b) + y^2) \)

Adding these two equations gives:
\[ 2y(a - b) = 0 \implies a = b \text{ or } y = 0 \]

### Step 7: Determine the Circle's Center
We know the locus of \( Z \) represents a circle with center \( \frac{3}{2} + \frac{i}{2} \). The general equation of a circle is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \( (h, k) \) is the center. Here, \( h = \frac{3}{2} \) and \( k = \frac{1}{2} \).

### Step 8: Relate to \( a \) and \( b \)
From the center, we have:
\[ a + b = 3 \quad \text{and} \quad a - b = 1 \]

### Step 9: Solve the System of Equations
Adding the two equations:
\[ 2a = 4 \implies a = 2 \]
Subtracting the second from the first:
\[ 2b = 2 \implies b = 1 \]

### Step 10: Calculate \( ab \)
Now, we find:
\[ ab = 2 \cdot 1 = 2 \]

### Final Answer
Thus, the value of \( ab \) is \( \boxed{2} \).