If `z_(1),z_(2) and z_(3)` are three distinct complex numbers such that `|z_(1)| = 1, |z_(2)| = 2, |z_(3)| = 4, arg(z_(2)) = arg(z_(1)) - pi, arg(z_(3)) = arg(z_(1)) + pi//2`, then `z_(2)z_(3)` is equal to

Options

A. `-8iz_(1)^(2)`
B. `8iz_(1)^(2)`
C. `-(8i)/(z_(i)^(2))`
D. `(8i)/(z_(i)^(2))`

Answer

To solve the problem, we need to find the product \( z_2 z_3 \) in terms of \( z_1 \). Let's break it down step by step:

### Step 1: Express \( z_1, z_2, z_3 \) in polar form
Given the magnitudes and arguments, we can express the complex numbers as follows:
- \( z_1 = |z_1| e^{i \arg(z_1)} = 1 e^{i \theta_1} = e^{i \theta_1} \)
- \( z_2 = |z_2| e^{i \arg(z_2)} = 2 e^{i (\theta_1 - \pi)} = 2 e^{i \theta_1} e^{-i \pi} = -2 e^{i \theta_1} \)
- \( z_3 = |z_3| e^{i \arg(z_3)} = 4 e^{i (\theta_1 + \frac{\pi}{2})} = 4 e^{i \theta_1} e^{i \frac{\pi}{2}} = 4 i e^{i \theta_1} \)

### Step 2: Calculate \( z_2 z_3 \)
Now, we can multiply \( z_2 \) and \( z_3 \):
\[ z_2 z_3 = (-2 e^{i \theta_1})(4 i e^{i \theta_1}) = -8 i e^{2 i \theta_1} \]

### Step 3: Express \( e^{2 i \theta_1} \) in terms of \( z_1 \)
Since \( z_1 = e^{i \theta_1} \), we have:
\[ e^{2 i \theta_1} = (e^{i \theta_1})^2 = z_1^2 \]
Thus, we can rewrite \( z_2 z_3 \):
\[ z_2 z_3 = -8 i z_1^2 \]

### Final Answer
The product \( z_2 z_3 \) is:
\[ z_2 z_3 = -8 i z_1^2 \]

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