Question

If \(|z_1| = 2, |z_2| = 3, |z_3| = 4\) and \(|2z_1 + 3z_2 + 4z_3| = 4\), then absolute value of \(8z_2z_3 + 27z_3z_1 + 64z_1z_2\) equals

• A. 24
• B. 48
• C. 72
• D. 96

Hint:

When you see expressions like \(k_1 z_2 z_3 + k_2 z_3 z_1 + k_3 z_1 z_2\), always try to factor out \(z_1 z_2 z_3\) and use the identity \(1/z = \bar{z}/|z|^2\). This often simplifies the problem to a known modulus given in the question.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The absolute value (modulus) of a complex number squared is equal to the product of the number and its conjugate (\( |z|^2 = z\bar{z} \)). We can use this property to rewrite the given expression in terms of the known modulus \( |2z_1 + 3z_2 + 4z_3| \).
Step 2: Key Formula or Approach:
1. \( z\bar{z} = |z|^2 \implies \bar{z} = \frac{|z|^2}{z} \)
2. \( |z_1 z_2 z_3| = |z_1||z_2||z_3| \)
Step 3: Detailed Explanation:
1. From the given magnitudes: \[ \bar{z}_1 = \frac{2^2}{z_1} = \frac{4}{z_1}, \quad \bar{z}_2 = \frac{3^2}{z_2} = \frac{9}{z_2}, \quad \bar{z}_3 = \frac{4^2}{z_3} = \frac{16}{z_3} \]
2. Consider the expression \( |8z_2z_3 + 27z_3z_1 + 64z_1z_2| \). Factor out \( z_1z_2z_3 \): \[ |z_1z_2z_3| \cdot \left| \frac{8}{z_1} + \frac{27}{z_2} + \frac{64}{z_3} \right| \]
3. Rewrite the terms inside the modulus using our conjugate substitutions: \[ (2 \cdot 3 \cdot 4) \cdot \left| 2\left(\frac{4}{z_1}\right) + 3\left(\frac{9}{z_2}\right) + 4\left(\frac{16}{z_3}\right) \right| \] \[ = 24 \cdot | 2\bar{z}_1 + 3\bar{z}_2 + 4\bar{z}_3 | \]
4. Since \( |\bar{z}| = |z| \), then \( |2\bar{z}_1 + 3\bar{z}_2 + 4\bar{z}_3| = |2z_1 + 3z_2 + 4z_3| \).
5. Substitute the given value \( |2z_1 + 3z_2 + 4z_3| = 4 \): \[ \text{Value} = 24 \times 4 = 96 \]
Step 4: Final Answer
The absolute value of the expression is 96.