Discussion
Let \(z_{1} = \frac{1}{2} + i\frac{\sqrt{3}}{2}\) and \(z_{2} = -\frac{1}{2} - i\frac{\sqrt{3}}{2}\) .If \(w = z_{1} + \overline{z_{2}}\) ,then \(\overline{w}\) =
Answer: D
\(\overline{w} =\) \(-i\sqrt{3}\).
Conjugating \(z_{2}\) changes the sign of its imaginary part:
\[ \begin{aligned} \overline{z_{2}} &= -\tfrac{1}{2} + i\tfrac{\sqrt{3}}{2} \\ w &= z_{1} + \overline{z_{2}} = i\sqrt{3} \\ \overline{w} &= -i\sqrt{3} \end{aligned} \]
The real parts cancel, so \(w\) is purely imaginary. \(i\sqrt{3}\) is \(w\) itself, not its conjugate; the real options \(\pm\sqrt{3}\) and 1 would need the imaginary parts to cancel instead.
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