4. A segment $A B(\overline{A B}=2 a)$ is given. Find the geometric locus of points $M$ for which $\overline{M A}^{2}-\overline{M B}^{2}=2 c(c-$ const $)$.
Solution. We choose a rectangular coordinate system such that the $x$-axis is the line $A B$, and the $y$-axis is the perpendicular bisector of the segment $A B$. Then we have $A(-a, 0)$, $B(a, 0)$. From the condition $\overline{M A}^{2}-\overline{M B}^{2}=2 c$, we get

$$
(x+a)^{2}+y^{2}-(x-a)^{2}-y^{2}=2 c
$$

From which we also get $x=\frac{c}{2 a}$. Therefore, the required geometric locus is a line

![](https://cdn.mathpix.com/cropped/2024_06_05_fb2f6c4e736d749cb214g-6.jpg?height=459&width=588&top_left_y=910&top_left_x=898)
perpendicular to the segment $A B$.