Example 5. Find the curve that has the property that the length of its arc, enclosed between any two points $P$ and $Q$, is proportional to the difference in the distances of points $P$ and $Q$ from a fixed point $O$.
Solution. If we fix point $P$, then the arc $Q P$ will change proportionally to the difference $O Q$ and the constant $O P$. Let's introduce polar coordinates,

![](https://cdn.mathpix.com/cropped/2024_05_22_a0612bb33256580e6dcdg-022.jpg?height=205&width=409&top_left_y=154&top_left_x=124)

Fig. 11

taking point $O$ as the pole and $O P$ as the polar axis (Fig. 11). The differential of the arc length of a curve in polar coordinates is

$$
(d s)^{2}=(d r)^{2}+(r d \varphi)^{2}
$$

From this, for our problem, we have

$$
k^{2}(d r)^{2}=(d r)^{2}+(r d \varphi)^{2}, \text { or } d \varphi=\sqrt{k^{2}-1} \frac{d r}{r}
$$

Integrating, we find $r=C e^{a \varphi}$ (logarithmic spiral).