Given an angle domain and a vector $\mathbf{v}$: Let $k$ be a circle that touches the boundary half-lines of the angle domain: $A$ and $B$ are two points on the circle such that $\overrightarrow{A B}=\mathbf{v}$. What is the geometric locus of the points $B$ if $k$ takes on all possible positions?
Let the opening angle of the sector be denoted by $2 \alpha$, and choose the positive direction of the $x$-axis of the coordinate system as the angle bisector of the sector. Let the unit be half the length of $\mathbf{v}$. Write the unit circle $k_{0}$ that is tangent to the sides of the sector, and let its center be $C_{0}$. Let $B_{0}$ be the endpoint of the vector $\overrightarrow{C_{0} B_{0}} = \frac{1}{2} \mathbf{v}$. $D_{0} D_{0}^{\prime}$ is the diameter perpendicular to $B_{0} C_{0}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_5599b44d37ebb11c6addg-1.jpg?height=892&width=981&top_left_y=362&top_left_x=561)

Any circle $k$ tangent to the sides can be obtained from $k_{0}$ by a dilation centered at $O$, with ratio $1: t$. The images of the points $C_{0}, D_{0}$, and $D_{0}^{\prime}$ are denoted by $C$, $D$, and $D^{\prime}$, respectively. For $B$, two points can come into question, their projections on $D D^{\prime}$ are $F$ and $F^{\prime}$, and

$$
C F = \sqrt{t^{2}-1} \cdot C D, \quad C F^{\prime} = \sqrt{t^{2}-1} \cdot C D^{\prime}
$$

holds. Therefore, the parametric equation of $B$ is

$$
\begin{gathered}
x = \cos \beta + \frac{t}{\sin \alpha} \pm \sqrt{t^{2}-1} \sin \beta \\
y = \sin \beta \mp \sqrt{t^{2}-1} \cos \beta
\end{gathered}
$$

where $\beta$ is the angle between the vector $\overrightarrow{C_{0} B_{0}}$ and the $x$-axis.

The obtained equation is the equation of the ray starting from $B_{0}$ and parallel to the positive $x$-axis if $\cos \beta = 0$, otherwise for the new variables

$$
\xi = \frac{x - \cos \beta}{\cos \beta}, \quad \eta = \frac{y - \sin \beta}{\cos \beta}
$$

we get the equation

$$
\xi \cos \beta = \frac{\sqrt{\eta^{2} + 1}}{\sin \alpha} - \eta \sin \beta
$$

which is the equation of one branch of a hyperbola.

Hasenfratz Anna, Budapest