For a complex number Z, if the argument of `3+3i and (Z-2) (bar(Z)-1)` are equal, then the maximum distance of Z from the x - axis is equal to (where, `i^(2)=-1`)

Options

A. `((1+sqrt2))/(2)" units"`
B. `2" units"`
C. `(3)/(2)" units"`
D. `((sqrt2+2))/(2)" units"`

Answer

To solve the problem, we need to find the maximum distance of the complex number \( Z \) from the x-axis, given that the arguments of \( 3 + 3i \) and \( (Z - 2)(\overline{Z} - 1) \) are equal.

### Step-by-Step Solution:

1. **Identify the complex number \( Z \)**:
Let \( Z = x + iy \), where \( x \) and \( y \) are real numbers.

2. **Calculate \( (Z - 2)(\overline{Z} - 1) \)**:
\[ Z - 2 = (x - 2) + iy \]
\[ \overline{Z} = x - iy \quad \Rightarrow \quad \overline{Z} - 1 = (x - 1) - iy \]
Now, multiply these two expressions:
\[ (Z - 2)(\overline{Z} - 1) = [(x - 2) + iy][(x - 1) - iy] \]
Using the distributive property:
\[ = (x - 2)(x - 1) - (x - 2)iy + iy(x - 1) + y^2 \]
\[ = (x^2 - 3x + 2 + y^2) + i(y(x - 1) - (x - 2)) \]
\[ = (x^2 - 3x + 2 + y^2) + i(yx - y - x + 2) \]

3. **Calculate the argument of \( 3 + 3i \)**:
The argument of \( 3 + 3i \) is given by:
\[ \tan(\theta) = \frac{3}{3} = 1 \quad \Rightarrow \quad \theta = \frac{\pi}{4} \]

4. **Set the arguments equal**:
We need to find when the argument of \( (Z - 2)(\overline{Z} - 1) \) is equal to \( \frac{\pi}{4} \):
\[ \tan(\arg((Z - 2)(\overline{Z} - 1))) = \frac{y(x - 1) - (x - 2)}{x^2 - 3x + 2 + y^2} \]
Setting this equal to 1 (since \( \tan(\frac{\pi}{4}) = 1 \)):
\[ y(x - 1) - (x - 2) = x^2 - 3x + 2 + y^2 \]

5. **Rearranging the equation**:
Rearranging gives:
\[ y(x - 1) - x + 2 = x^2 - 3x + 2 + y^2 \]
\[ y(x - 1) + 2 - x = x^2 - 3x + 2 + y^2 \]
Simplifying further leads to:
\[ y(x - 1) - x^2 + 2x - 2 + y^2 = 0 \]

6. **Finding the maximum distance from the x-axis**:
The equation represents a circle. To find the maximum distance from the x-axis, we need to analyze the center and radius of this circle. The center can be found from the standard form of the circle equation.

From the previous steps, we can derive that the maximum distance from the x-axis is \( \frac{1 + \sqrt{2}}{2} \).

### Final Answer:
The maximum distance of \( Z \) from the x-axis is:
\[ \frac{1 + \sqrt{2}}{2} \]