Let `Z_(1)=x_(1)+iy_(1)`, `Z_(2)=x_(2)+iy_(2)` be complex numbers in fourth quadrant of argand plane and `|Z_(1)|=|Z_(2)|=1`, `Ref(Z_(1)Z_(2))=0`. The complex numbers `Z_(3)=x_(1)+ix_(2)`, `Z_(4)=y_(1)+iy_(2)`, `Z_(5)=x_(1)+iy_(2)`, `Z_(6)=x_(6)+iy`, will always satisfy

Options

A. `|Z_(4)|=1`
B. `arg(Z_(1)Z_(4))=-pi//2`
C. `(Z_(5))/(cos(argZ_(1)))+(Z_(6))/(sin(argZ_(1)))` is purely real
D. `Z_(5)^(2)+(barZ_(6))^(2)` is purely imaginergy

Explanation

To solve the problem, we will analyze the given complex numbers and their properties step by step.

### Step 1: Understand the given complex numbers
Let \( Z_1 = x_1 + iy_1 \) and \( Z_2 = x_2 + iy_2 \) be complex numbers in the fourth quadrant of the Argand plane. This means:
- \( x_1 > 0 \) (real part is positive)
- \( y_1 < 0 \) (imaginary part is negative)
- \( x_2 > 0 \)
- \( y_2 < 0 \)

### Step 2: Modulus of the complex numbers
We know that \( |Z_1| = |Z_2| = 1 \). Therefore:
\[ |Z_1| = \sqrt{x_1^2 + y_1^2} = 1 \implies x_1^2 + y_1^2 = 1 \]
\[ |Z_2| = \sqrt{x_2^2 + y_2^2} = 1 \implies x_2^2 + y_2^2 = 1 \]

### Step 3: Argument condition
Given that \( \text{Re}(Z_1 Z_2) = 0 \), we can express this in terms of the arguments of the complex numbers. Let:
\[ Z_1 = e^{i\theta_1}, \quad Z_2 = e^{i\theta_2} \]
where \( \theta_1 \) and \( \theta_2 \) are the arguments of \( Z_1 \) and \( Z_2 \) respectively. Since \( Z_1 \) and \( Z_2 \) are in the fourth quadrant:
\[ -\frac{\pi}{2} < \theta_1 < 0, \quad -\frac{\pi}{2} < \theta_2 < 0 \]
The condition \( \text{Re}(Z_1 Z_2) = 0 \) implies:
\[ \theta_1 + \theta_2 = -\frac{\pi}{2} \]

### Step 4: Define the new complex numbers
Now we define the new complex numbers:
- \( Z_3 = x_1 + ix_2 \)
- \( Z_4 = y_1 + iy_2 \)
- \( Z_5 = x_1 + iy_2 \)
- \( Z_6 = x_2 + iy \) (assuming \( y \) is a variable)

### Step 5: Analyze \( Z_3 \), \( Z_4 \), \( Z_5 \), and \( Z_6 \)
1. **For \( Z_3 \)**:
\[ Z_3 = x_1 + ix_2 \]
The real part is \( x_1 \) (positive) and the imaginary part is \( x_2 \) (positive), thus \( Z_3 \) is in the first quadrant.

2. **For \( Z_4 \)**:
\[ Z_4 = y_1 + iy_2 \]
The real part is \( y_1 \) (negative) and the imaginary part is \( y_2 \) (negative), thus \( Z_4 \) is in the third quadrant.

3. **For \( Z_5 \)**:
\[ Z_5 = x_1 + iy_2 \]
The real part is \( x_1 \) (positive) and the imaginary part is \( y_2 \) (negative), thus \( Z_5 \) is in the fourth quadrant.

4. **For \( Z_6 \)**:
\[ Z_6 = x_2 + iy \]
The real part is \( x_2 \) (positive) and the imaginary part is \( y \) (variable). Depending on the value of \( y \), \( Z_6 \) can be in the first or fourth quadrant.

### Conclusion
The complex numbers \( Z_3 \), \( Z_4 \), \( Z_5 \), and \( Z_6 \) will always satisfy the conditions based on their definitions and the properties of \( Z_1 \) and \( Z_2 \).