4A. What does the geometric place of points in the plane Oxy equidistant from the circle $x^{2}+y^{2}=1$ and the point $A(3,0)$ represent?
Solution. Let $M(x, y)$ be an arbitrary point of the desired geometric locus and $N$ be the point where the line $O M$ (where $O$ is the origin) intersects the circle. Since $\overline{N M}=\overline{A M}$, we have

$$
\begin{aligned}
& \overline{N M}=\overline{O M}-\overline{O N}=\overline{O M}-1 \\
& \overline{O M}=\sqrt{x^{2}+y^{2}} \\
& \overline{A M}=\sqrt{(x-3)^{2}+y^{2}}
\end{aligned}
$$

we get

$$
\sqrt{x^{2}+y^{2}}-1=\sqrt{(x-3)^{2}+y^{2}}
$$

By squaring, we obtain

$$
\sqrt{x^{2}+y^{2}}=3 x-4
$$

![](https://cdn.mathpix.com/cropped/2024_06_05_56aeac1763a3f1e8ce2ag-09.jpg?height=367&width=533&top_left_y=1244&top_left_x=935)

By squaring again, we obtain

$$
\begin{aligned}
& 8 x^{2}-24 x-y^{2}+16=0 \\
& 8\left(x-\frac{3}{2}\right)^{2}-y^{2}=2
\end{aligned}
$$

and this is the equation of a hyperbola.