3B. Find the geometric place of points in the plane for which the squares of the distances from a given point and from a given line are constant.
Solution. Let point $A$ not lie on the line $l$. In the plane, we choose a coordinate system such that the $x$-axis coincides with the line $l$, and point $A$ lies on the $y$-axis. Point $A$ has coordinates $\left(0, y_{0}\right), y_{0} \neq 0$.

Let $M(x, y)$ be an arbitrary point from the desired geometric locus and let $M M_{1} \perp l, M_{1}(x, 0)$. By the condition:

$$
\overline{M A}^{2}-{\overline{M M_{1}}}^{2}=c, \quad c \in \mathbb{R}
$$

From this, we find:

$$
\begin{aligned}
& x^{2}+\left(y-y_{0}\right)^{2}-y^{2}=c \\
& x^{2}+y_{0}^{2}-2 y y_{0}=c \\
& y=\frac{1}{2 y_{0}} x^{2}+\frac{y_{0}^{2}-c}{2 y_{0}}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_06_05_bd3f37f65d5d6e754f63g-13.jpg?height=336&width=447&top_left_y=803&top_left_x=1019)

Thus, the desired geometric locus is a parabola. In this case: