1. Given real numbers $a, b$ satisfy
$$
\left(a-\sqrt{a^{2}-1}\right)\left(b-\sqrt{b^{2}-1}\right)=1 \text {. }
$$

Statements: (1) $a=b$; (2) $a+b=0$; (3) $a b=1$; (4) $a b=-1$. Among these, the correct number of statements is $(\quad)$.
(A) 1
(B) 2
(C) 3
(D) 0
-,1. B.
Given that multiplying both sides of the equation by $a+\sqrt{a^{2}-1}$ yields
$$
b-\sqrt{b^{2}-1}=a+\sqrt{a^{2}-1} \text {. }
$$

Similarly, $b+\sqrt{b^{2}-1}=a-\sqrt{a^{2}-1}$.
(1) + (2) gives $a=b$.
$$
\begin{array}{l}
\text { Hence }\left(a-\sqrt{a^{2}-1}\right)^{2}=1 \Rightarrow a= \pm 1 \\
\Rightarrow a b=a^{2}=1 .
\end{array}
$$