Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = a + bi$, where $a$ and $b$ are real numbers, the modulus is defined as $|z| = \sqrt{a^2 + b^2}$.
• Unimodular Complex Number: A complex number $z$ is unimodular if $|z| = 1$.
• Properties of Modulus and Conjugates:
  • $|z|^2 = z\overline{z}$, where $\overline{z}$ is the complex conjugate of $z$.
  • $|z_1/z_2| = |z_1|/|z_2|$
  • $\overline{z_1 - z_2} = \overline{z_1} - \overline{z_2}$
  • $\overline{z_1 z_2} = \overline{z_1} \overline{z_2}$
  • $\overline{\overline{z}} = z$

Step-by-Step Solution

1. Express the given condition mathematically

We are given that $\frac{z_1 - 2z_2}{2 - z_1\overline{z_2}}$ is unimodular. This means its modulus is equal to 1. $\left| \frac{z_1 - 2z_2}{2 - z_1\overline{z_2}} \right| = 1$ Explanation: We directly translate the problem's premise into a mathematical equation using the definition of unimodularity.

2. Apply the modulus property for division

Using the property $\left| \frac{A}{B} \right| = \frac{|A|}{|B|}$, we can rewrite the equation as: $\frac{|z_1 - 2z_2|}{|2 - z_1\overline{z_2}|} = 1$ This implies: $|z_1 - 2z_2| = |2 - z_1\overline{z_2}|$ Explanation: If the ratio of two moduli is 1, it means the moduli themselves are equal.

3. Square both sides

To eliminate the modulus signs, we square both sides of the equation: $|z_1 - 2z_2|^2 = |2 - z_1\overline{z_2}|^2$ Explanation: Squaring both sides allows us to use the identity $|z|^2 = z\overline{z}$, which simplifies the expression by removing the modulus.

4. Apply the identity $|z|^2 = z\overline{z}$

Using the identity $|z|^2 = z\overline{z}$, we have: $(z_1 - 2z_2)(\overline{z_1 - 2z_2}) = (2 - z_1\overline{z_2})(\overline{2 - z_1\overline{z_2}})$ Applying the conjugate properties $\overline{A-B} = \overline{A} - \overline{B}$ and $\overline{AB} = \overline{A}\overline{B}$, and $\overline{\overline{z}} = z$ we get: $(z_1 - 2z_2)(\overline{z_1} - 2\overline{z_2}) = (2 - z_1\overline{z_2})(2 - \overline{z_1}z_2)$ Explanation: We replace the squared moduli with products of complex numbers and their conjugates, making the equation easier to manipulate algebraically.

5. Expand both sides

Expanding both sides of the equation: Left Hand Side (LHS): $z_1\overline{z_1} - 2z_1\overline{z_2} - 2z_2\overline{z_1} + 4z_2\overline{z_2}$ Using $z\overline{z} = |z|^2$: $|z_1|^2 - 2z_1\overline{z_2} - 2\overline{z_1}z_2 + 4|z_2|^2$ Right Hand Side (RHS): $4 - 2z_1\overline{z_2} - 2\overline{z_1}z_2 + z_1\overline{z_2}\overline{z_1}z_2$ Using $z\overline{z} = |z|^2$: $4 - 2z_1\overline{z_2} - 2\overline{z_1}z_2 + |z_1|^2|z_2|^2$ Equating LHS and RHS: $|z_1|^2 - 2z_1\overline{z_2} - 2\overline{z_1}z_2 + 4|z_2|^2 = 4 - 2z_1\overline{z_2} - 2\overline{z_1}z_2 + |z_1|^2|z_2|^2$ Explanation: We perform algebraic expansion and use the identity $|z|^2 = z\overline{z}$.

6. Simplify the equation

After canceling common terms, the equation becomes: $|z_1|^2 + 4|z_2|^2 = 4 + |z_1|^2|z_2|^2$ Rearranging the terms: $|z_1|^2 - |z_1|^2|z_2|^2 - 4 + 4|z_2|^2 = 0$ Explanation: We simplify the equation by canceling out identical terms on both sides.

7. Factor the equation

Factoring by grouping: $|z_1|^2(1 - |z_2|^2) - 4(1 - |z_2|^2) = 0$ $(|z_1|^2 - 4)(1 - |z_2|^2) = 0$ Explanation: We factor the equation to isolate $|z_1|$.

8. Use the condition that $z_2$ is not unimodular

Since $z_2$ is not unimodular, $|z_2| \ne 1$, which means $1 - |z_2|^2 \ne 0$. Therefore, we must have: $|z_1|^2 - 4 = 0$ $|z_1|^2 = 4$ Taking the square root: $|z_1| = 2$ Explanation: Using the given information that $z_2$ is not unimodular, we can determine that $(1 - |z_2|^2)$ is not zero. Thus, the other factor must be zero, which allows us to solve for $|z_1|$.

9. Geometric interpretation

The equation $|z_1| = 2$ represents a circle centered at the origin with a radius of 2 in the complex plane. Explanation: We interpret the algebraic result geometrically.

Conclusion: The point $z_1$ lies on a circle of radius 2.

Common Mistakes & Tips

• Careless Conjugate Expansion: Be extra careful when expanding conjugate expressions. Double-check your application of $\overline{AB} = \overline{A}\overline{B}$ and $\overline{A+B} = \overline{A} + \overline{B}$.
• Forgetting $z_2$ is NOT Unimodular: This condition is essential. Without it, you can't proceed to isolate $|z_1|$.
• Not Squaring Correctly: Squaring both sides is a standard trick, but make sure you expand $(a+b)^2$ and $(a-b)^2$ correctly.

Summary

We used the definition of a unimodular complex number, along with properties of moduli and conjugates, to establish a relationship between $|z_1|$ and $|z_2|$. The key insight was using the fact that $z_2$ is not unimodular to conclude that $|z_1|^2 = 4$, leading to $|z_1| = 2$. This corresponds to the locus of $z_1$ being a circle centered at the origin with radius 2.

Final Answer

The final answer is \boxed{circle of radius 2}, which corresponds to option (A).