Question

Let $z_1, z_2$, and $z_3$ be complex numbers satisfying the following conditions:
\[ 2 = |2z_1| = |z_2 - 1| = |z_3 + 1| = \left| \frac{1}{z_1} + \frac{1}{z_2 - 1} + \frac{1}{z_3 + 1} \right| \]
What is the value of $|4z_1 + z_2 + z_3|$?

• A. 8
• B. 4
• C. $\frac{1}{4}$
• D. $\frac{1}{8}$

Hint:

For any complex equation involving sums of reciprocals $\frac{1}{w}$, always think of substituting $\frac{1}{w} = \frac{\bar{w}}{|w|^2}$.
This is a standard technique that converts reciprocals into direct linear terms of conjugates, which are much easier to simplify.

Answer 1

The Correct Option is A

Step 1 : Understanding the Question:
The question asks us to find the modulus of the complex expression $|4z_1 + z_2 + z_3|$ given the moduli of individual terms and the modulus of the sum of their reciprocals.

Step 2 : Key Formulas and Approach:
For any complex number $w$, we have the property:
\[ |w|^2 = w \bar{w} \implies \frac{1}{w} = \frac{\bar{w}}{|w|^2} \]
We will use this identity to rewrite the reciprocals of the complex terms in the given equation and then simplify the resulting expression.

Step 3 : Detailed Explanation:
Let us analyze the individual given moduli first:

• $|2z_1| = 2 \implies |z_1| = 1 \implies |z_1|^2 = 1$

• $|z_2 - 1| = 2 \implies |z_2 - 1|^2 = 4$

• $|z_3 + 1| = 2 \implies |z_3 + 1|^2 = 4$
Using the reciprocal property $\frac{1}{w} = \frac{\bar{w}}{|w|^2}$ for each term:

• For $z_1$:
\[ \frac{1}{z_1} = \frac{\bar{z}_1}{|z_1|^2} = \bar{z}_1 \]

• For $z_2 - 1$:
\[ \frac{1}{z_2 - 1} = \frac{\bar{z}_2 - 1}{|z_2 - 1|^2} = \frac{\bar{z}_2 - 1}{4} \]

• For $z_3 + 1$:
\[ \frac{1}{z_3 + 1} = \frac{\bar{z}_3 + 1}{|z_3 + 1|^2} = \frac{\bar{z}_3 + 1}{4} \]
Substitute these expressions into the given sum modulus equation:
\[ \left| \frac{1}{z_1} + \frac{1}{z_2 - 1} + \frac{1}{z_3 + 1} \right| = 2 \]
\[ \left| \bar{z}_1 + \frac{\bar{z}_2 - 1}{4} + \frac{\bar{z}_3 + 1}{4} \right| = 2 \]
Let us factor out $\frac{1}{4}$ from the expression inside the modulus:
\[ \left| \frac{4\bar{z}_1 + (\bar{z}_2 - 1) + (\bar{z}_3 + 1)}{4} \right| = 2 \]
Using the property $|k w| = |k||w|$:
\[ \frac{1}{4} \left| 4\bar{z}_1 + \bar{z}_2 + \bar{z}_3 \right| = 2 \]
\[ \left| \overline{4z_1 + z_2 + z_3} \right| = 8 \]
Since the modulus of a complex conjugate is equal to the modulus of the complex number itself ($|\bar{u}| = |u|$):
\[ |4z_1 + z_2 + z_3| = 8 \]

Step 4 : Final Answer:
The value of $|4z_1 + z_2 + z_3|$ is 8.
This matches Option (A).