7.8. Given two non-intersecting circles. Find the geometric locus of the centers of circles that bisect the given circles (i.e., intersect them at diametrically opposite points).
7.8. Let $O_{1}$ and $O_{2}$ be the centers of the given circles, and $R_{1}$ and $R_{2}$ their radii. A circle of radius $r$ with center $X$ intersects the first circle at diametrically opposite points if and only if $r^{2}=X O_{1}^{2}+R_{1}^{2}$, therefore the desired locus of points consists of points $X$ such that $X O_{1}^{2}+R_{1}^{2}=X O_{2}^{2}+R_{2}^{2}$, all such points $X$ lie on a line perpendicular to $O_{1} O_{2}$ (Problem 7.6).