For any two complex numbers `z_(1),z_(2)` the values of `|z_(1)+z_(2)|^(2)+|z_(1)-z_(2)|^(2)`, is

Options

A. `|z_(1)|^(2)+|z_(2)|^(2)`
B. `2(|z_(1)|^(2)+|z_(2)|^(2))`
C. `(|z_(1)|+|z_(2)|)^(2)`
D. none of these

Answer

To solve the problem, we need to find the value of \( |z_1 + z_2|^2 + |z_1 - z_2|^2 \) for any two complex numbers \( z_1 \) and \( z_2 \).

### Step-by-step Solution:

1. **Define the Complex Numbers**:
Let \( z_1 = a + ib \) and \( z_2 = x + iy \), where \( a, b, x, y \) are real numbers.

2. **Calculate \( |z_1 + z_2|^2 \)**:
\[ z_1 + z_2 = (a + x) + i(b + y) \]
The modulus squared is:
\[ |z_1 + z_2|^2 = (a + x)^2 + (b + y)^2 \]

3. **Calculate \( |z_1 - z_2|^2 \)**:
\[ z_1 - z_2 = (a - x) + i(b - y) \]
The modulus squared is:
\[ |z_1 - z_2|^2 = (a - x)^2 + (b - y)^2 \]

4. **Combine the Two Results**:
Now, we need to add these two results:
\[ |z_1 + z_2|^2 + |z_1 - z_2|^2 = ((a + x)^2 + (b + y)^2) + ((a - x)^2 + (b - y)^2) \]

5. **Expand the Squares**:
Expanding both squares:
\[ (a + x)^2 = a^2 + 2ax + x^2 \]
\[ (b + y)^2 = b^2 + 2by + y^2 \]
\[ (a - x)^2 = a^2 - 2ax + x^2 \]
\[ (b - y)^2 = b^2 - 2by + y^2 \]

6. **Combine All Terms**:
Now, substituting back:
\[ |z_1 + z_2|^2 + |z_1 - z_2|^2 = (a^2 + 2ax + x^2 + b^2 + 2by + y^2) + (a^2 - 2ax + x^2 + b^2 - 2by + y^2) \]
Combine like terms:
\[ = 2a^2 + 2b^2 + 2x^2 + 2y^2 \]

7. **Factor Out the Common Factor**:
\[ = 2(a^2 + b^2 + x^2 + y^2) \]

8. **Relate to Modulus**:
Recognizing that:
\[ a^2 + b^2 = |z_1|^2 \quad \text{and} \quad x^2 + y^2 = |z_2|^2 \]
We can write:
\[ = 2(|z_1|^2 + |z_2|^2) \]

### Final Answer:
Thus, the value of \( |z_1 + z_2|^2 + |z_1 - z_2|^2 \) is:
\[ \boxed{2(|z_1|^2 + |z_2|^2)} \]