## Task 17/65

Determine the geometric locus for the center of a sphere with a given radius $r$ that touches two intersecting lines at an angle of $2 \alpha$.
Preliminary Consideration: The center of the sphere has the same distance from the straight lines that are tangent to the sphere in every position. Therefore, the geometric locus of the sphere's center belongs to the two symmetry planes of the pair of lines.

These planes are perpendicular to the plane determined by the two lines and intersect this plane along the two angle bisectors.

![](https://cdn.mathpix.com/cropped/2024_06_06_1d2be4837fb839c2b31dg-115.jpg?height=463&width=916&top_left_y=1088&top_left_x=570)

The geometric locus of the center of a sphere that touches only one straight line is a cylindrical surface whose radius is equal to the sphere's radius and whose axis is the line.

The sought locus is thus the intersection of the two symmetry planes with the cylinder of radius \( r \) around one of the two lines. Since a non-parallel plane intersects a cylinder in an ellipse, the geometric locus consists of two ellipses whose minor axes coincide and whose major axes are perpendicular to each other.

The semi-minor axes are equal to the sphere's radius \( r \), and the semi-major axes are derived from the diagram (which represents the ground plan of the intersection) as

\[
\frac{r}{\sin \alpha} \quad \text{and} \quad \frac{r}{\cos \alpha}
\]