Eight, on one side $0 x$ of the acute angle $x o y$, there is a point $\mathrm{A}$, from which a perpendicular line is drawn to the other side, with its foot of the perpendicular being $\mathrm{B}$, then from $\mathrm{B}$ a perpendicular line is drawn to $0 \mathrm{x}$, with the foot of the perpendicular being $\mathrm{C}$, and this process is repeated infinitely. Find the total length of all the perpendicular lines (where $\mathrm{AB}=$
$$
\mathrm{a}, \mathrm{BC}=\mathrm{b} \text { ). }
$$
Eight, solve $\because \triangle A B C \backsim \triangle B C D \backsim \triangle C D E \sim \triangle D E F \backsim$
$$
\begin{array}{l}
\therefore \frac{B C}{A B}=\frac{b}{a}=\frac{C D}{B C}=\frac{D E}{C D}=\frac{E F}{D E}=\cdots \\
C D=B C \times \frac{b}{a}=b \times \frac{b}{a}, \\
\mathrm{DE}=\mathrm{CD} \times \frac{b}{a}=b \times\left(\frac{b}{a}\right)^{2}, \\
E F=D E \times \frac{b}{a}=b \times\left(\frac{b}{a}\right)^{3},
\end{array}
$$

The total length of the segments
$$
\begin{aligned}
S & =A B+B C+C D+D E+E F+\cdots \\
& =a+b+b \times \frac{b}{a}+b \times\left(\frac{b}{a}\right)^{2}+b \times
\end{aligned}
$$
$$
\left(\frac{b}{a}\right)^{3}+\cdots
$$

According to the problem, $\mathrm{a}>\mathrm{b}>0$,
we have $0<\frac{b}{a}<1$,
$$
\therefore S=a+\frac{b}{1-\frac{b}{a}}=\frac{a^{2}}{a-b} \text {. }
$$