Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = x + iy$, the modulus is $|z| = \sqrt{x^2 + y^2}$, representing the distance from the origin to the point $(x, y)$ in the complex plane. $|z - c|$ represents the distance between $z$ and $c$ in the complex plane.
• Argument of a Complex Number: For a complex number $z = x + iy$, the argument, $\arg(z)$, is the angle $\theta$ that the vector from the origin to $(x, y)$ makes with the positive real axis, such that $-\pi < \theta \le \pi$. We can find $\theta$ using $\tan \theta = \frac{y}{x}$.

Step-by-Step Solution

1. Representing the Complex Numbers

• Why this step: To manipulate the given conditions algebraically, we represent the complex numbers in terms of their real and imaginary parts. Let $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$, where $x_1, y_1, x_2, y_2$ are real numbers. Our goal is to find $\text{Im}(z_1 + z_2) = y_1 + y_2$.

2. Interpreting the First Given Condition: $\text{Re}(z) = |z - 1|$

• Why this step: We translate the given condition into an algebraic equation involving the real and imaginary parts of the complex numbers. For $z_1$, we have $\text{Re}(z_1) = |z_1 - 1|$. Substituting $z_1 = x_1 + iy_1$: $x_1 = |(x_1 + iy_1) - 1|$ $x_1 = |(x_1 - 1) + iy_1|$ Applying the definition of the modulus: $x_1 = \sqrt{(x_1 - 1)^2 + y_1^2}$ Squaring both sides (note that $x_1 \ge 0$): $x_1^2 = (x_1 - 1)^2 + y_1^2$ $x_1^2 = x_1^2 - 2x_1 + 1 + y_1^2$ $0 = -2x_1 + 1 + y_1^2$ Rearranging the terms: $y_1^2 = 2x_1 - 1 \quad (*)$ This equation represents a parabola opening to the right with vertex at $(\frac{1}{2}, 0)$.

Similarly, for $z_2$, we have $\text{Re}(z_2) = |z_2 - 1|$.

• Why this step: Since the same condition applies to $z_2$, we obtain a similar equation for its real and imaginary parts. Following the same steps: $x_2 = |(x_2 - 1) + iy_2|$ $x_2^2 = (x_2 - 1)^2 + y_2^2$ $x_2^2 = x_2^2 - 2x_2 + 1 + y_2^2$ $y_2^2 = 2x_2 - 1 \quad (**)$

3. Combining the Conditions for $z_1$ and $z_2$

• Why this step: We subtract the equations for $z_1$ and $z_2$ to eliminate the constant term and establish a relationship between $x_1, y_1, x_2,$ and $y_2$ that can be used with the argument condition. Subtract equation $(**)$ from equation $(*)$: $(y_1^2) - (y_2^2) = (2x_1 - 1) - (2x_2 - 1)$ $y_1^2 - y_2^2 = 2x_1 - 2x_2$ Factoring the difference of squares: $(y_1 - y_2)(y_1 + y_2) = 2(x_1 - x_2)$
• Important Note: This step is valid as long as $y_1 \ne y_2$. If $y_1 = y_2$, then from $(y_1 - y_2)(y_1 + y_2) = 2(x_1 - x_2)$, it would imply $2(x_1 - x_2) = 0$, so $x_1 = x_2$. This means $z_1 = z_2$, which would make $\arg(z_1 - z_2)$ undefined. Hence, $y_1 \ne y_2$ is assured by the argument condition. Isolating $(y_1 + y_2)$: $y_1 + y_2 = 2 \left( \frac{x_1 - x_2}{y_1 - y_2} \right) \quad (***)$

4. Interpreting the Second Given Condition: $\arg(z_1 - z_2) = \frac{\pi}{6}$

• Why this step: The argument condition provides a direct relationship between the real and imaginary parts of $(z_1 - z_2)$. This will give us the ratio $\frac{y_1 - y_2}{x_1 - x_2}$. First, find the difference $z_1 - z_2$: $z_1 - z_2 = (x_1 + iy_1) - (x_2 + iy_2)$ $z_1 - z_2 = (x_1 - x_2) + i(y_1 - y_2)$ The argument of a complex number $X + iY$ is $\tan^{-1}\left(\frac{Y}{X}\right)$. So, for $z_1 - z_2$: $\arg(z_1 - z_2) = \tan^{-1}\left(\frac{y_1 - y_2}{x_1 - x_2}\right)$ We are given that $\arg(z_1 - z_2) = \frac{\pi}{6}$: $\tan^{-1}\left(\frac{y_1 - y_2}{x_1 - x_2}\right) = \frac{\pi}{6}$ Take the tangent of both sides: $\frac{y_1 - y_2}{x_1 - x_2} = \tan\left(\frac{\pi}{6}\right)$ We know that $\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}$. Therefore: $\frac{y_1 - y_2}{x_1 - x_2} = \frac{1}{\sqrt{3}} \quad (****)$

5. Final Calculation

• Why this step: We need to find $y_1 + y_2$. The expression in () contains the inverse of the ratio found in (**). By substituting, we can directly calculate $y_1 + y_2$. From (): $\frac{x_1 - x_2}{y_1 - y_2} = \sqrt{3}$ Substitute this into (): $y_1 + y_2 = 2 \left( \sqrt{3} \right)$ $y_1 + y_2 = 2\sqrt{3}$ Finally, we need to find $\text{Im}(z_1 + z_2)$. $z_1 + z_2 = (x_1 + iy_1) + (x_2 + iy_2)$ $z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2)$ So, $\text{Im}(z_1 + z_2) = y_1 + y_2$. Therefore: $\text{Im}(z_1 + z_2) = 2\sqrt{3}$

Since the problem states that the correct answer is (A) $\frac{\sqrt{3}}{2}$ and our calculation gives $2\sqrt{3}$, there is an error in the problem statement, the options, or the provided correct answer. However, following the strict instructions, the answer must be the derived value of $2\sqrt{3}$.

Common Mistakes & Tips

• Carefully track signs during algebraic manipulations, especially when subtracting equations or expanding squared terms.
• Remember that $x_1 = \text{Re}(z_1) = |z_1 - 1|$ implies $x_1 \ge 0$.
• Note that $\arg(z_1 - z_2) = \frac{\pi}{6}$ implies $z_1 \neq z_2$.
• When substituting, make sure you're using the correct ratio ($\frac{x_1 - x_2}{y_1 - y_2}$ or $\frac{y_1 - y_2}{x_1 - x_2}$).

Summary

We represented the complex numbers $z_1$ and $z_2$ in terms of their real and imaginary parts. We used the condition $\text{Re}(z) = |z - 1|$ to derive equations for the locus of $z_1$ and $z_2$ as parabolas. Subtracting the equations allowed us to relate $y_1 + y_2$ to the ratio of $x_1 - x_2$ and $y_1 - y_2$. Finally, the argument condition $\arg(z_1 - z_2) = \frac{\pi}{6}$ provided the value of this ratio, enabling us to calculate $\text{Im}(z_1 + z_2) = y_1 + y_2$.

The final answer is \boxed{2\sqrt{3}}.