5. Satisfy $a>b>2$ and make $a b$, $\frac{b}{a}$, $a-b$, $a+b$ can be arranged in a certain order to form a geometric sequence of real number pairs $(a, b)(\quad)$.
(A) does not exist
(B) exactly 1
(C) exactly 2
(D) infinitely many
5. (B).

From the problem, we know that $a b$, $\frac{b}{a}$, $a+b$, and $a-b$ are all positive numbers and $a b > a$.
$$
\begin{array}{l}
+b> \frac{b}{a}, a b > a + b > a - b . \\
\because (a+b)^{2} \neq a b \cdot \frac{b}{a},
\end{array}
$$
$\therefore$ The geometric sequence can only be $a b$, $a+b$, $a-b$, $\frac{b}{a}$.
$$
\therefore\left\{\begin{array}{l}
(a+b)^{2}=a b(a-b), \\
(a+b)(a-b)=a b \cdot \frac{b}{a} .
\end{array}\right.
$$

Thus, $a=7+5 \sqrt{2}, b=\frac{1}{2}(10+7 \sqrt{2})$.