[ [MT - line or segment]

On the plane, points $A$ and $B$ are given. Find the locus of points $M$ for which the difference of the squares of the lengths of segments $A M$ and $B M$ is constant.

#
Let's introduce a coordinate system, choosing point $A$ as the origin and directing the $O x$ axis along the ray $A B$. Let point $M$ have coordinates $(x, y)$. Then $A M^{2}=x^{2}+y^{2}$ and $B M^{2}=(x-a)^{2}+y^{2}$, where $a=A B$. Therefore, $A M^{2}-B M^{2}=$ $2 a x-a^{2}$. This value equals $k$ for points $M$ with coordinates $\left(\left(a^{2}+k\right) / 2 a, y\right)$; all such points lie on a line perpendicular to $A B$.