4B. Find the geometric place of points in a plane for which the difference of the squares of the distances from a given point and from a given line is constant and equal to s.
Solution. Let A be the given point and suppose it does not lie on the given line $l$. We choose a coordinate system with the $x$-axis as the given line, and the $y$-axis as the normal to the given line passing through $A$. Then $A\left(0, y_{0}\right)$.

Let $M(x, y)$ be an arbitrary point of the desired geometric locus and $M_{1}(x, 0)$ be the projection of $M$ onto the $x$-axis. Then $\overline{A M}^{2}+{\overline{M_{1} M}}^{2}=c$, i.e.,

$$
x^{2}+\left(y-y_{0}\right)^{2}-y^{2}=c
$$

From this,

![](https://cdn.mathpix.com/cropped/2024_06_05_84a5520b68919fd51f4cg-12.jpg?height=346&width=450&top_left_y=788&top_left_x=1015)

$$
2 y y_{0}=x^{2}+y_{0}^{2}-c
$$

1) If $y_{0} \neq 0$, i.e., $A$ does not lie on $l$, the desired geometric locus is the parabola $y=\frac{1}{2 y_{0}} x^{2}+\frac{y_{0}^{2}-c}{2 y_{0}}$.
2) If $y_{0}=0$, i.e., $A$ lies on $l$, we get $x^{2}=c$, so the desired geometric locus is the lines $x=\sqrt{c}$ and $x=-\sqrt{c}$ for $c \geq 0$, and the empty set for $c<0$.