Let four points `z_(1),z_(2),z_(3),z_(4)` be in complex plane such that `|z_(2)|= 1,` `|z_(1)|leq 1` and `|z_(3)| le 1`. If `z_(3) = (z_(2)(z_(1)-z_(4)))/(barz_(1)z_(4)-1)`, then `|z_(4)|` can be

Options

A. 2
B. `(2)/(5)`
C. `(1)/(3)`
D. `(5)/(2)`

Explanation

To solve the problem, we need to analyze the given conditions and the expression for \( z_3 \).

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- We have four complex numbers \( z_1, z_2, z_3, z_4 \).
- The conditions are:
- \( |z_2| = 1 \)
- \( |z_1| \leq 1 \)
- \( |z_3| \leq 1 \)
- The relationship given is:
\[ z_3 = \frac{z_2(z_1 - z_4)}{\overline{z_1} z_4 - 1} \]

2. **Finding the Modulus of \( z_3 \):**
- We need to find \( |z_3| \) using the modulus properties.
- Taking the modulus of both sides, we have:
\[ |z_3| = \left| \frac{z_2(z_1 - z_4)}{\overline{z_1} z_4 - 1} \right| \]
- Using the property \( |a/b| = |a|/|b| \):
\[ |z_3| = \frac{|z_2| \cdot |z_1 - z_4|}{|\overline{z_1} z_4 - 1|} \]

3. **Substituting Known Values:**
- Since \( |z_2| = 1 \):
\[ |z_3| = \frac{|z_1 - z_4|}{|\overline{z_1} z_4 - 1|} \]

4. **Using the Condition for \( |z_3| \):**
- We know \( |z_3| \leq 1 \):
\[ \frac{|z_1 - z_4|}{|\overline{z_1} z_4 - 1|} \leq 1 \]
- This implies:
\[ |z_1 - z_4| \leq |\overline{z_1} z_4 - 1| \]

5. **Analyzing the Inequality:**
- The inequality \( |z_1 - z_4| \leq |\overline{z_1} z_4 - 1| \) gives us a relationship between \( z_1 \) and \( z_4 \).
- We can analyze this further by substituting \( z_4 = re^{i\theta} \) where \( r = |z_4| \) and \( \theta \) is the argument of \( z_4 \).

6. **Finding the Maximum Modulus of \( z_4 \):**
- Since \( |z_1| \leq 1 \) and \( |z_3| \leq 1 \), we can conclude that \( |z_4| \) must also be constrained.
- By analyzing the limits, we find that \( |z_4| \) can take values up to 1.

### Conclusion:
Thus, the maximum value of \( |z_4| \) is 1, and it can be:

\[ |z_4| \leq 1 \]