(12 分) 复数 $z_{1}$ 所对应的点在点 $(1,1)$ 及 $(1,-1)$ 为端点的线段上运动, 复数 $z_{2}$满足 $\left|z_{2}\right|=1$, 求:

(1) 复数 $z_{1} \cdot z_{2}$ 模的取值范围;

(2) 复数 $z_{1}^{2}$ 对应的点的轨迹方程.
(1) 设 $z_{1}=1-t i,(t \in[-1,1]), z_{2}=x-y i,\left(x^{2}+y^{2}=1
ight)$,则 $\left|z_{1} z_{2}
ight|=|(1-t i) \cdot(x-y i)|=|(x-y t)-(t x+y) \mathrm{i}|=\sqrt{1+t^{2}} \in[1, \sqrt{2}]$(2) $z_{1}^{2}=1-t^{2}-2 t i \Rightarrow\left\{egin{array}{c}x=1-t^{2} \ u=2 t\end{array} 
ightarrow y^{2}-4 x-4=0(-2 \leq y \leq 1,0 \leq x \leq 1)
ight.$.