A Partition and a Parallelogram

Sets $A,B,C$ with $A\cap B=\varnothing$ and $A\cup B=C$ are said to give a partition $(A,B)$ of $C$.

1. Let $A_1=\left\{x\mid\tan\left(\dfrac{\pi x}2+\dfrac{\pi}4\right)=-1,\ x\in\mathbb R\right\}$, $B_1=\{x\mid\cos(\pi x)=1,\ x\in\mathbb R\}$, $C_1=\{x\mid\sin(\pi x)=0,\ x\in\mathbb R\}$. Determine whether $(A_1,B_1)$ is a partition of $C_1$, and justify. 2. Let $f(x)=\sqrt{\dfrac{x-a}{x-b}}$ ($a>b$) and

Let $A_2=\{x\mid y=f(x)\}$ (the domain of $f$) and $B_2=\{y\mid y=g(x)\}$ (the range of $g$). Given that when $\lambda=5$, $\mu=4$, the pair $(A_2,B_2)$ is a partition of $\mathbb R$: a parallelogram $P_1P_2P_3P_4$ has all four vertices on the graph of $h(x)=\log_2\dfrac{x+1}{x-1}$, with the $x$-coordinate of $P_1$ equal to $a-7$ and that of $P_2$ equal to $-\dfrac 23b$. Find the area of the parallelogram $P_1P_2P_3P_4$.