40. Find the geometric locus of points $M$ on a plane, the difference of the squares of the distances from which to two given points $A$ and $B$ on this plane is constant and equal to $a$. For which values of $a$ does the problem have solutions?
Solution. Introduce a rectangular coordinate system as shown in Figure 18, where point $O$ is the midpoint of segment $A B$. Let $M(x ; y)$ be the point of the desired set. By the condition $A M^{2}-B M^{2}=a$, or (considering $A B=2 c$)

$$
(x+c)^{2}+y^{2}-\left((x-c)^{2}+y^{2}\right)=C, \quad \text { or } \quad x=\frac{a}{4 c} .
$$

In the obtained formula, the parameter $a$ takes any values, and this formula defines a set of lines parallel to the axis $O y$, and the axis itself.