Question

If \( |z_1| = 2, |z_2| = 3, |z_3| = 4 \) and \( |2z_1 + 3z_2 + 4z_3| = 4 \), then the absolute value of \( 8z_2z_3 + 27z_1z_3 + 64z_1z_2 \) equals:

• A. 24
• B. 48
• C. 72
• D. 96

Hint:

For problems involving absolute values of complex expressions, use the properties of magnitudes and simplify each term before combining them.

Answer 1

The Correct Option is D

We are given the equation \( |8z_2z_3 + 27z_1z_3 + 64z_1z_2| = |z_1| |z_2| |z_3| \). First, we break down the terms as follows: \[ \left| \frac{8}{z_1} + \frac{27}{z_2} + \frac{64}{z_3} \right| \] This can be rewritten as: \[ = (2)(3)(4) \left| \frac{8z_1}{|z_1|^2} + \frac{27z_2}{|z_2|^2} + \frac{64z_3}{|z_3|^2} \right| \] Simplifying the equation: \[ = 24 \left| 2\overline{z_1} + 3\overline{z_2} + 4\overline{z_3} \right| \] Finally, we calculate: \[ = 24 \left| 2z_1 + 3z_2 + 4z_3 \right| \] This results in: \[ = 24 \times 4 = 96 \] Thus, the final answer is 96.