5. Let circles $O_{1}$ and $O_{2}$ be two fixed circles, and let moving circle $P$ be tangent to both of these fixed circles. Then, the locus of the center of circle $P$ cannot be

保留源文本的换行和格式，直接输出翻译结果如下：

5. Let circles $O_{1}$ and $O_{2}$ be two fixed circles, and let moving circle $P$ be tangent to both of these fixed circles. Then, the locus of the center of circle $P$ cannot be
Let the radii of circles $O_{1}$ and $O_{2}$ be $r_{1}$ and $r_{2}$ ($r_{1} \geqslant r_{2}$). The trajectory of point $P$ has the following four cases:
(1) $\left|P O_{1}\right|+\left|P O_{2}\right|=r_{1}+r_{2}$;
(2) $||P O_{1}|-| P O_{2}||=r_{1}+r_{2}$;
(3) $\left|P O_{1}\right|+\left|P O_{2}\right|=r_{1}-r_{2}$;
(4) $||P O_{1}|-| P O_{2}||=r_{1}-r_{2}$.

When circles $O_{1}$ and $O_{2}$ are separate, the trajectory is a hyperbola (2) (tangent to one circle externally and the other internally) and a hyperbola (4) (tangent to both circles externally or internally), corresponding to figure $D$; if $r_{1}=r_{2}$, the trajectory is a hyperbola (2) (tangent to one circle externally and the other internally) and the perpendicular bisector of segment $O_{1} O_{2}$, corresponding to figure $B$;
When circles $O_{1}$ and $O_{2}$ are externally tangent, the trajectory is the line $O_{1} O_{2}$ and a hyperbola (4) (tangent to both circles externally or internally); if $r_{1}=r_{2}$, the trajectory is the line $O_{1} O_{2}$ and the perpendicular bisector of segment $O_{1} O_{2}$; when circles $O_{1}$ and $O_{2}$ intersect, the trajectory is an ellipse (1) (tangent to one circle externally and the other internally) and a hyperbola (4) (tangent to both circles externally or internally);
When circles $O_{1}$ and $O_{2}$ are internally tangent, the trajectory is an ellipse (1) (tangent to one circle externally and the other internally) and the line $O_{1} O_{2}$; when circles $O_{1}$ and $O_{2}$ are nested, the trajectory is an ellipse (1) (tangent to one circle externally and the other internally) and an ellipse (3) (tangent to both circles internally), corresponding to figure $C$.
Since in figure $A$, the foci of the hyperbola and the ellipse do not coincide, it cannot occur. Therefore, the answer is $A$.