If `z_(1) and z_(2)` are complex number with `|z_(1)|-|z_(2)|,` then which of the following is/are correct?
1.`z_(1)=z_(2)`
2. Real part of `z_(1)=` Real part of `z_(2)`
3. Imaginary part of `z_(1)=` Imaginary part of `z_(2)`
Select the correct answer using the code given below :

Options

A. 1 only
B. 2 only
C. 3 only
D. None

Explanation

To solve the problem, we need to analyze the given information about the complex numbers \( z_1 \) and \( z_2 \). The condition provided is that the modulus of \( z_1 \) is equal to the modulus of \( z_2 \), i.e.,

\[ |z_1| = |z_2| \]

### Step 1: Understanding the Modulus of Complex Numbers
Let us express the complex numbers in terms of their real and imaginary parts:
- \( z_1 = a_1 + i b_1 \)
- \( z_2 = a_2 + i b_2 \)

The modulus of a complex number \( z = a + ib \) is given by:

\[ |z| = \sqrt{a^2 + b^2} \]

Thus, we have:

\[ |z_1| = \sqrt{a_1^2 + b_1^2} \]
\[ |z_2| = \sqrt{a_2^2 + b_2^2} \]

### Step 2: Setting Up the Equation
Given that \( |z_1| = |z_2| \), we can write:

\[ \sqrt{a_1^2 + b_1^2} = \sqrt{a_2^2 + b_2^2} \]

Squaring both sides gives:

\[ a_1^2 + b_1^2 = a_2^2 + b_2^2 \]

### Step 3: Analyzing the Statements
Now, let's analyze the three statements given in the question:

1. **Statement 1**: \( z_1 = z_2 \)
- This implies \( a_1 + ib_1 = a_2 + ib_2 \), which means \( a_1 = a_2 \) and \( b_1 = b_2 \). However, the equality of moduli does not guarantee that the real and imaginary parts are equal. Therefore, this statement is **false**.

2. **Statement 2**: Real part of \( z_1 = \) Real part of \( z_2 \)
- This means \( a_1 = a_2 \). Again, the equality of moduli does not imply that the real parts are equal. Therefore, this statement is also **false**.

3. **Statement 3**: Imaginary part of \( z_1 = \) Imaginary part of \( z_2 \)
- This means \( b_1 = b_2 \). Similar to the previous statements, the equality of moduli does not imply that the imaginary parts are equal. Therefore, this statement is also **false**.

### Conclusion
Since all three statements are false, the correct answer is that none of the statements are correct.

### Final Answer
The correct answer is: **None of the statements are correct.**