2. Two circles are given which touch each other and have radii $r$ and $2 r$. Find the geometric locus of points equally distant from the given circles.
Solution. Let's place the given circles in a rectangular coordinate system such that their centers lie on the $O x$-axis, and the point of tangency is at the coordinate origin.

Case I. The circles touch each other externally (see diagram).

If $T(x, y)$ is a point outside the circles, which belongs to the desired geometric locus, then from the condition $\overline{T T_{1}}=\overline{T_{2}}$, i.e., $\overline{T O_{1}}-2 r=\overline{\mathrm{TO}_{2}}-r$, we obtain

$$
8\left(x+\frac{r}{2}\right)^{2}-y^{2}=2 r^{2}
$$

which represents the equation of a hyperbola. It is evident that all points on the segment $\mathrm{O}_{1} \mathrm{O}_{2}$ also belong to the geometric locus.

Case II. The circles touch each other internally (see diagram).
![](https://cdn.mathpix.com/cropped/2024_06_05_3d1981dfa5d34b5b59cfg-7.jpg?height=482&width=1174&top_left_y=1210&top_left_x=266)

In this case, it is also evident that all points on the non-positive part of the $O x$-axis belong to the desired geometric locus of points. Are there any other points in this geometric locus? If $T(x, y)$ is such a point, then it is clear that $T$ must be outside the smaller circle and inside the larger circle. Then, from the condition of the problem, we get $\overline{T T_{1}}=\overline{T T_{2}}$, i.e., $2 r-\overline{T O_{1}}=\overline{T O_{2}}-r$, from which we obtain

$$
8\left(x-\frac{3 r}{2}\right)^{2}+9 y^{2}=18 r^{2}
$$

which represents the equation of an ellipse.