For any two complex numbers `z_(1)` and `z_(2)`
`|z_(1)+z_(2)|^(2) =(|z_(1)|^(2)+|z_(2)|^(2))`

Options

A. Statement-1 is True, Statement-2 is True: Statement-2 is a correct exp,anation for statement-1.
B. Statement-1 is true, statement -2 is true, Statement-2 is not a correct explanation for statement-1.
C. Statement-1 is True, statement-2 is false,
D. statement-1 is False, Statement-2 is true.

Answer

To solve the problem, we need to prove that for any two complex numbers \( z_1 \) and \( z_2 \), the equation

\[ |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 \]

holds true under certain conditions. Let's go through the steps to derive the solution.

### Step 1: Start with the left-hand side
We begin with the left-hand side of the equation:

\[ |z_1 + z_2|^2 \]

Using the property of modulus, we can express this as:

\[ |z_1 + z_2|^2 = (z_1 + z_2)(\overline{z_1 + z_2}) \]

### Step 2: Expand the expression
Now, we expand the expression:

\[ (z_1 + z_2)(\overline{z_1} + \overline{z_2}) = z_1\overline{z_1} + z_1\overline{z_2} + z_2\overline{z_1} + z_2\overline{z_2} \]

### Step 3: Substitute the modulus
Recall that \( |z_1|^2 = z_1\overline{z_1} \) and \( |z_2|^2 = z_2\overline{z_2} \). Thus, we can rewrite the expression as:

\[ |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + z_1\overline{z_2} + z_2\overline{z_1} \]

### Step 4: Recognize the real part
The terms \( z_1\overline{z_2} + z_2\overline{z_1} \) can be rewritten as:

\[ 2 \text{Re}(z_1\overline{z_2}) \]

So we have:

\[ |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + 2 \text{Re}(z_1\overline{z_2}) \]

### Step 5: Set the equation
According to the problem statement, we have:

\[ |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 \]

This implies:

\[ |z_1|^2 + |z_2|^2 + 2 \text{Re}(z_1\overline{z_2}) = |z_1|^2 + |z_2|^2 \]

### Step 6: Simplify the equation
Subtract \( |z_1|^2 + |z_2|^2 \) from both sides:

\[ 2 \text{Re}(z_1\overline{z_2}) = 0 \]

### Step 7: Conclusion
This means that:

\[ \text{Re}(z_1\overline{z_2}) = 0 \]

This indicates that \( z_1 \) and \( z_2 \) are orthogonal in the complex plane, which implies that \( z_1 \) and \( z_2 \) are perpendicular to each other.

### Final Result
Thus, the condition for the given equation to hold is:

\[ \text{Re}\left(\frac{z_1}{z_2}\right) = 0 \]

This means that the real part of the quotient of \( z_1 \) and \( z_2 \) is zero.

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