Let `z_(1), z_(2), z_(3)` be three complex numbers such that `|z_(1)| = |z_(2)| = |z_(3)| = 1 and z = (z_(1) + z_(2) + z_(3))((1)/(z_(1))+(1)/(z_(2))+(1)/(z_(3)))`, then |z| cannot exceed

Options

A. 1
B. 3
C. 6
D. 9

Answer

To solve the problem, we need to find the maximum value of the modulus of the complex number \( z \) defined as:

\[ z = (z_1 + z_2 + z_3) \left( \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right) \]

Given that \( |z_1| = |z_2| = |z_3| = 1 \), we can proceed with the following steps:

### Step 1: Rewrite the expression for \( z \)

We can rewrite the second part of the expression using the property of complex conjugates. Since \( |z_i| = 1 \), we have:

\[ \frac{1}{z_i} = \overline{z_i} \]

Thus, we can express \( z \) as:

\[ z = (z_1 + z_2 + z_3) \left( \overline{z_1} + \overline{z_2} + \overline{z_3} \right) \]

### Step 2: Apply the modulus

Now, we take the modulus of \( z \):

\[ |z| = |(z_1 + z_2 + z_3)(\overline{z_1} + \overline{z_2} + \overline{z_3})| \]

Using the property of modulus, we can separate the terms:

\[ |z| = |z_1 + z_2 + z_3| \cdot |\overline{z_1} + \overline{z_2} + \overline{z_3}| \]

Since the modulus of a complex number and its conjugate are equal, we have:

\[ |z| = |z_1 + z_2 + z_3| \cdot |z_1 + z_2 + z_3| = |z_1 + z_2 + z_3|^2 \]

### Step 3: Use the triangle inequality

By the triangle inequality, we know that:

\[ |z_1 + z_2 + z_3| \leq |z_1| + |z_2| + |z_3| = 1 + 1 + 1 = 3 \]

Thus, we can square this result:

\[ |z| \leq (|z_1 + z_2 + z_3|)^2 \leq 3^2 = 9 \]

### Step 4: Conclusion

Therefore, we conclude that the maximum value of \( |z| \) cannot exceed 9:

\[ |z| \leq 9 \]