8,9

Given a circle $S$ and a point $M$ outside it. Through the point $M$, all possible circles $S_{1}$ intersecting the circle $S$ are drawn; $X$ is the intersection point of the tangent at point $M$ to the circle $S_{1}$ with the extension of the common chord of circles $S$ and $S_{1}$. Find the locus of $X$.

#
Let $A$ and $B$ be the points of intersection of the circles $S$ and $S_{1}$. Then $X M^{2}=X A \cdot X B=X O^{2}-R^{2}$, where $O$ and $R$ are the center and radius of the circle $S$. Therefore, $X O^{2}-X M^{2}=R^{2}$, which means that the points $X$ lie on the perpendicular to the line $O M$ (see problem 7.6).