Question

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

• A. 24
• B. 48
• C. 72
• D. 96

Hint:

Whenever you see a sum of products like \( z_1z_2 + z_2z_3 + z_3z_1 \) paired with individual magnitudes, try factoring out \( z_1z_2z_3 \). It often transforms the expression into a sum of conjugates which is much easier to handle.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The problem utilizes properties of complex numbers, specifically the relationship \( |z|^2 = z \bar{z} \). We can factor out the product of the three complex numbers from the expression we need to evaluate.

Step 2: Key Formula or Approach:
Use \( \bar{z} = \frac{|z|^2}{z} \). For the given magnitudes: \[ \bar{z}_1 = \frac{4}{z_1}, \quad \bar{z}_2 = \frac{9}{z_2}, \quad \bar{z}_3 = \frac{16}{z_3} \]

Step 3: Detailed Explanation:
Let \( S = 8z_2 z_3 + 27z_3 z_1 + 64z_1 z_2 \). Factor out \( z_1 z_2 z_3 \): \[ S = z_1 z_2 z_3 \left( \frac{8}{z_1} + \frac{27}{z_2} + \frac{64}{z_3} \right) \] We can rewrite the constants using the magnitudes: \[ S = z_1 z_2 z_3 \left( 2 \cdot \frac{4}{z_1} + 3 \cdot \frac{9}{z_2} + 4 \cdot \frac{16}{z_3} \right) \] \[ S = z_1 z_2 z_3 ( 2\bar{z}_1 + 3\bar{z}_2 + 4\bar{z}_3 ) \] Taking the absolute value: \[ |S| = |z_1| |z_2| |z_3| |2\bar{z}_1 + 3\bar{z}_2 + 4\bar{z}_3| \] Since \( |\bar{Z}| = |Z| \), we have \( |2\bar{z}_1 + 3\bar{z}_2 + 4\bar{z}_3| = |2z_1 + 3z_2 + 4z_3| = 4 \). \[ |S| = (2)(3)(4)(4) = 96 \]

Step 4: Final Answer
The absolute value of the expression is 96.