Let `x_(1) and y_(1)` be real number. If `z_(1) and z_(2)` are complex numbers such that `|z_(1)|=|z_(2)|=4`, then `|x_(1)z_(1)-y_(1)z_(2)|^(2)+|y_(1)z_(1)+x_(1)z_(2)|^(2)=`

Options

A. `32(x_(1)^(2)+y_(1)^(2))`
B. `16(x_(1)^(2)+y_(1)^(2))`
C. `4(x_(1)^(2)+y_(1)^(2))`
D. `32(x_(1)^(2)+y_(1)^(2))|z_(1)+z_(2)|^(2)`

Answer

To solve the problem, we need to evaluate the expression \( |x_1 z_1 - y_1 z_2|^2 + |y_1 z_1 + x_1 z_2|^2 \) given that \( |z_1| = |z_2| = 4 \).

### Step-by-Step Solution:

1. **Understanding the Modulus**:
Since \( |z_1| = |z_2| = 4 \), we can express this in terms of their squares:
\[ |z_1|^2 = 16 \quad \text{and} \quad |z_2|^2 = 16. \]

2. **Expanding the First Term**:
We start with the first term \( |x_1 z_1 - y_1 z_2|^2 \):
\[ |x_1 z_1 - y_1 z_2|^2 = |x_1 z_1|^2 + |y_1 z_2|^2 - 2 \text{Re}(x_1 z_1 \overline{(y_1 z_2)}). \]
Here, \( |x_1 z_1|^2 = x_1^2 |z_1|^2 = x_1^2 \cdot 16 = 16 x_1^2 \) and \( |y_1 z_2|^2 = y_1^2 |z_2|^2 = y_1^2 \cdot 16 = 16 y_1^2 \).

3. **Calculating the Cross Term**:
The cross term becomes:
\[ -2 \text{Re}(x_1 z_1 \overline{(y_1 z_2)}) = -2 x_1 y_1 \text{Re}(z_1 \overline{z_2}). \]

4. **Expanding the Second Term**:
Now, we expand the second term \( |y_1 z_1 + x_1 z_2|^2 \):
\[ |y_1 z_1 + x_1 z_2|^2 = |y_1 z_1|^2 + |x_1 z_2|^2 + 2 \text{Re}(y_1 z_1 \overline{(x_1 z_2)}). \]
Similarly, we have \( |y_1 z_1|^2 = 16 y_1^2 \) and \( |x_1 z_2|^2 = 16 x_1^2 \).

5. **Calculating the Cross Term for the Second Term**:
The cross term here is:
\[ 2 \text{Re}(y_1 z_1 \overline{(x_1 z_2)}) = 2 y_1 x_1 \text{Re}(z_1 \overline{z_2}). \]

6. **Combining Both Terms**:
Now we combine both expanded terms:
\[ |x_1 z_1 - y_1 z_2|^2 + |y_1 z_1 + x_1 z_2|^2 = (16 x_1^2 + 16 y_1^2 - 2 x_1 y_1 \text{Re}(z_1 \overline{z_2})) + (16 y_1^2 + 16 x_1^2 + 2 y_1 x_1 \text{Re}(z_1 \overline{z_2})). \]

7. **Simplifying the Expression**:
The terms involving \( \text{Re}(z_1 \overline{z_2}) \) will cancel out:
\[ = 32 x_1^2 + 32 y_1^2. \]

8. **Final Result**:
Thus, we conclude:
\[ |x_1 z_1 - y_1 z_2|^2 + |y_1 z_1 + x_1 z_2|^2 = 32 (x_1^2 + y_1^2). \]

### Final Answer:
\[ |x_1 z_1 - y_1 z_2|^2 + |y_1 z_1 + x_1 z_2|^2 = 32 (x_1^2 + y_1^2). \]