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

The given equation is \( |8z_2z_3 + 27z_1z_3 + 64z_1z_2| = |z_1| |z_2| |z_3| \). We analyze the term \(\left| \frac{8}{z_1} + \frac{27}{z_2} + \frac{64}{z_3} \right|\). This can be expressed 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, we get \(24 \left| 2\overline{z_1} + 3\overline{z_2} + 4\overline{z_3} \right|\). Further simplification leads to \(24 \left| 2z_1 + 3z_2 + 4z_3 \right|\). The final calculation is \(24 \times 4 = 96\). The result is 96.