設 \(O\) 為複數平面上的原點,並令點 \(A, B\) 分別代表非零複數 \(z, w\)。若 \(\angle AOB = 90^\circ\),則下列哪些選項必為負實數？
(1) \(\frac{z}{w}\)
(2) \(zw\)
(3) \((zw)^2\)
(4) \(\frac{z^2}{w^2}\)
(5) \((z\overline{w})^2\) （其中 \(\overline{w}\) 為 \(w\) 的共軛複數）
\[
\boxed{\text{已知條件}}
\]
\[
\begin{aligned}
z &= a(\cos\theta + i\sin\theta),\ a>0 \\
w &= b(\cos\alpha + i\sin\alpha),\ b>0 \\
\theta - \alpha &= \pm 90^\circ
\end{aligned}
\]
\[
\boxed{\text{關鍵計算}}
\]
\[
\begin{aligned}
\frac{z}{w} &= \frac{a}{b} \big[\cos(\theta-\alpha) + i\sin(\theta-\alpha)\big] = \frac{a}{b}(\pm i) \\
\frac{z^2}{w^2} &= \frac{a^2}{b^2} \big[\cos(2\theta-2\alpha) + i\sin(2\theta-2\alpha)\big] \\
&= \frac{a^2}{b^2} \cos(\pm 180^\circ) = -\frac{a^2}{b^2} < 0 \\
(zw)^2 &= a^2 b^2 \big[\cos(2\theta-2\alpha) + i\sin(2\theta-2\alpha)\big] \\
&= -a^2 b^2 < 0
\end{aligned}
\]
\[
\boxed{\text{選項判斷}}
\]
\[
\begin{array}{c|c}
\text{選項} & \text{判斷與理由} \\ \hline
(1)\ z/w & \text{純虛數（不恆正負）} \Rightarrow \times \\
(2)\ zw & \text{無法確定正負} \Rightarrow \times \\
(3)\ \text{同(2)} & \times \\
(4)\ z^2/w^2 & \text{恆負實數} \Rightarrow \bigcirc \\
(5)\ (zw)^2 & \text{恆負實數} \Rightarrow \bigcirc
\end{array}
\]
\[
\therefore \text{答案：}(4)(5)
\]