If `z_(1)` and `z_(2)` are complex numbers with `| z_(1)| = | z_(2) |`, then which of the following is `//` are correct ?
I. `z_(1) = z_(2)`
II. Real part of `z_(1) =` real part of `z_(2)`
III. Imaginary part of `z_(1) = `Imaginary part of `z_(2)`.
Select the correct answer using the code given below

Options

A. Only I
B. Only II
C. Only III
D. None of these

Answer

To solve the problem, we need to analyze the statements given the condition that the magnitudes of two complex numbers \( z_1 \) and \( z_2 \) are equal, i.e., \( |z_1| = |z_2| \).

### Step-by-Step Solution:

1. **Understanding the Magnitude of Complex Numbers:**
The magnitude of a complex number \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \), where \( a \) is the real part and \( b \) is the imaginary part.

2. **Given Condition:**
We are given that \( |z_1| = |z_2| \). This means:
\[ \sqrt{a_1^2 + b_1^2} = \sqrt{a_2^2 + b_2^2} \]
where \( z_1 = a_1 + b_1 i \) and \( z_2 = a_2 + b_2 i \).

3. **Analyzing the Statements:**
- **Statement I:** \( z_1 = z_2 \)
- This statement is **not necessarily true**. Two complex numbers can have the same magnitude but be different in value. For example, \( z_1 = 1 + 3i \) and \( z_2 = -1 - 3i \) both have the same magnitude \( \sqrt{10} \) but are not equal.
- **Conclusion:** Statement I is **false**.

- **Statement II:** Real part of \( z_1 = \) real part of \( z_2 \)
- This statement is also **not necessarily true**. The real parts can be different while the magnitudes are the same. For instance, in the previous example, the real parts are \( 1 \) and \( -1 \), which are not equal.
- **Conclusion:** Statement II is **false**.

- **Statement III:** Imaginary part of \( z_1 = \) imaginary part of \( z_2 \)
- Similar to the previous statements, this statement is **not necessarily true**. The imaginary parts can also differ while the magnitudes are the same. Again, using the previous example, the imaginary parts are \( 3 \) and \( -3 \), which are not equal.
- **Conclusion:** Statement III is **false**.

4. **Final Conclusion:**
Since all three statements are false, the correct answer is that none of the statements are true.

### Answer:
None of the statements (I, II, III) are correct.