If $C$ is a circle in the $3$-dimensional euclidean space there is a $1$-parameter family of spheres $S_k$ that contain $C$.
If $C_1$ and $C_2$ are two circles which are not complanar and intersect in two distinct points, does it always exist a sphere $S$ that contains both circles?
For a single circle $C$, a sphere containing $C$ is uniquely determined by its center, which lie on the line $\Delta_C$ going through the center of $C$ and orthogonal to the plane containing $C$.
Let $A,B$ be the points at which the two circles $C_1,C_2$ intersect. Let $P$ be the bisector plane of the segment $[AB]$. Then $\Delta_{C_1}$ and $\Delta_{C_2}$ both lie in $P$. Since $C_1$ and $C_2$ are not coplanar, $\Delta_{C_1}$ and $\Delta_{C_2}$ are not parallel. Therefore, they intersect at a point $D$. There is a sphere with center $D$ and containing both $C_1$ and $C_2$.