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 (a) `2` (b) `2/5` (c) `1/3` (d) `5/2`

Options

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

Explanation

To solve the problem, we will analyze the given equation and the conditions step by step.

### Given:
1. \( |z_2| = 1 \)
2. \( |z_1| \leq 1 \)
3. \( |z_3| \leq 1 \)
4. \( z_3 = \frac{z_2(z_1 - z_4)}{\overline{z_1} z_4 - 1} \)

### Step 1: Taking the modulus of both sides
We start by taking the modulus of both sides of the equation for \( z_3 \):
\[ |z_3| = \left| \frac{z_2(z_1 - z_4)}{\overline{z_1} z_4 - 1} \right| \]

### Step 2: Applying properties of modulus
Using the properties of modulus, we can rewrite this as:
\[ |z_3| = \frac{|z_2| \cdot |z_1 - z_4|}{|\overline{z_1} z_4 - 1|} \]

### Step 3: Substituting known values
Since \( |z_2| = 1 \), we have:
\[ |z_3| = \frac{|z_1 - z_4|}{|\overline{z_1} z_4 - 1|} \]

### Step 4: Using the condition for \( z_3 \)
Given that \( |z_3| \leq 1 \), we can write:
\[ \frac{|z_1 - z_4|}{|\overline{z_1} z_4 - 1|} \leq 1 \]

### Step 5: Rearranging the inequality
This implies:
\[ |z_1 - z_4| \leq |\overline{z_1} z_4 - 1| \]

### Step 6: Squaring both sides
To eliminate the modulus, we square both sides:
\[ |z_1 - z_4|^2 \leq |\overline{z_1} z_4 - 1|^2 \]

### Step 7: Expanding both sides
Expanding both sides gives:
\[ |z_1|^2 + |z_4|^2 - 2 \text{Re}(z_1 \overline{z_4}) \leq |z_1|^2 |z_4|^2 - 2 \text{Re}(\overline{z_1} z_4) + 1 \]

### Step 8: Simplifying the inequality
Rearranging terms leads to:
\[ |z_4|^2 - 2 \text{Re}(z_1 \overline{z_4}) + 2 \text{Re}(\overline{z_1} z_4) - 1 \leq 0 \]

### Step 9: Analyzing the conditions
Since \( |z_1| \leq 1 \), we can analyze the implications for \( |z_4| \). The inequality suggests that \( |z_4| \) must be constrained.

### Step 10: Conclusion
From the analysis, we find that \( |z_4| \) must be less than or equal to 1 to satisfy the conditions given. The possible values for \( |z_4| \) from the options provided are:
- (a) \( 2 \)
- (b) \( \frac{2}{5} \)
- (c) \( \frac{1}{3} \)
- (d) \( \frac{5}{2} \)

Among these, the only feasible value that satisfies \( |z_4| \leq 1 \) is \( \frac{2}{5} \) and \( \frac{1}{3} \).

Thus, the answer is:
**(b) \( \frac{2}{5} \)** or **(c) \( \frac{1}{3} \)**.