For any two complex numbers, `z_(1),z_(2)` and any two real numbers a and b, `|az_(1)-bz_(2)|^(2)+|bz_(1)+az_(2)|^(2)=`

Options

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

Answer

To solve the expression \( |az_1 - bz_2|^2 + |bz_1 + az_2|^2 \), we will break it down step by step.

### Step 1: Define the expressions
Let:
- \( A = |az_1 - bz_2|^2 \)
- \( B = |bz_1 + az_2|^2 \)

### Step 2: Expand \( A \)
Using the property of magnitudes, we have:
\[ A = |az_1 - bz_2|^2 = (az_1 - bz_2)(az_1 - bz_2)^* \]
where \( z^* \) denotes the complex conjugate of \( z \).

Now, we can express the conjugate:
\[ (az_1 - bz_2)^* = a z_1^* - b z_2^* \]
Thus,
\[ A = (az_1 - bz_2)(a z_1^* - b z_2^*) \]

### Step 3: Multiply the terms
Expanding the product:
\[ A = a^2 z_1 z_1^* - ab z_1 z_2^* - ab z_2 z_1^* + b^2 z_2 z_2^* \]
Using the property \( z z^* = |z|^2 \), we can rewrite this as:
\[ A = a^2 |z_1|^2 + b^2 |z_2|^2 - ab(z_1 z_2^* + z_2 z_1^*) \]

### Step 4: Expand \( B \)
Now, we expand \( B \):
\[ B = |bz_1 + az_2|^2 = (bz_1 + az_2)(bz_1 + az_2)^* \]
The conjugate is:
\[ (bz_1 + az_2)^* = b z_1^* + a z_2^* \]
Thus,
\[ B = (bz_1 + az_2)(b z_1^* + a z_2^*) \]

### Step 5: Multiply the terms
Expanding this product:
\[ B = b^2 z_1 z_1^* + ab z_1 z_2^* + ab z_2 z_1^* + a^2 z_2 z_2^* \]
Again using \( z z^* = |z|^2 \):
\[ B = b^2 |z_1|^2 + a^2 |z_2|^2 + ab(z_1 z_2^* + z_2 z_1^*) \]

### Step 6: Combine \( A \) and \( B \)
Now we combine \( A \) and \( B \):
\[ A + B = (a^2 |z_1|^2 + b^2 |z_2|^2 - ab(z_1 z_2^* + z_2 z_1^*)) + (b^2 |z_1|^2 + a^2 |z_2|^2 + ab(z_1 z_2^* + z_2 z_1^*)) \]

### Step 7: Simplify the expression
Combining like terms:
\[ A + B = (a^2 + b^2)|z_1|^2 + (a^2 + b^2)|z_2|^2 \]
This simplifies to:
\[ A + B = (a^2 + b^2)(|z_1|^2 + |z_2|^2) \]

### Final Result
Thus, the final result is:
\[ |az_1 - bz_2|^2 + |bz_1 + az_2|^2 = (a^2 + b^2)(|z_1|^2 + |z_2|^2) \]