89. Pursuit (II). Ship $O_{1}$ notices another ship $O_{2}$, which at the moment of observation is moving perpendicular to the line $O_{1} O_{2}$. Ship $O_{2}$ does not notice the signals from $O_{1}$ and maintains its original course and speed $v_{2}$. Ship $O_{1}$ wants to attract attention, as it needs help, and with its maximum speed $v_{1}$, which it is capable of, it heads in such a direction to get as close as possible to $O_{2}$. What course should it take? What will be the distance between the two ships at the moment of closest approach, if the initial distance was $d$, and the ratio of speeds $v_{1} / v_{2}$ is $k$ and less than 1?
89. Let $P_{1}, P_{2}$ and $O_{1}$, $O_{2}$ denote the positions of both ships at the initial moment (when the second ship is first sighted by the first) and at the moment when their distance is the smallest. If so, the first ship sailed from point $P_{1}$ to $O_{1}$ along the straight line $P_{1} O_{2}$ (Fig. 164). Moreover, at the final moment, the components of the velocities of both ships along $O_{1} O_{2}$ must be equal, i.e.,

$$
v_{1}=v_{2} \sin \alpha, \sin \alpha=k
$$

The angle $\alpha$ determines the course of the first ship. Since the ships reached points $O_{1}, O_{2}$ at the same time, we have

$$
\frac{P_{1} O_{1}}{P_{2} O_{2}}=\frac{v_{1}}{v_{2}}=\sin \alpha
$$

Since

$$
P_{2} O_{2}=d \operatorname{tg} \alpha, \quad P_{1} O_{2}=\frac{d}{\cos \alpha}
$$

the sought minimum distance between the ships is

$$
\text { }
$$

$$
\begin{aligned}
O_{1} O_{2} & =\frac{d}{\cos \alpha}-\frac{d \sin ^{2} \alpha}{\cos \alpha}= \\
& =d \cos \alpha=d \sqrt{1-k^{2}}
\end{aligned}
$$