3. Find the geometric locus of points $P(u, v)$ which are equally distant from the curve $x^{2}+4 x+y^{2}=0$ and the point $M(2,0)$.
Solution. The curve $x^{2}+4 x+y^{2}=0$ is a circle with center $S(-2,0)$ and radius 2 (see diagram). If $N(x, y)$ is any point on the circle, then according to the condition of the problem, we have $\overline{N P}=\overline{M P}, \quad \overline{S P}=\overline{S N}+\overline{N P}=2+\overline{M P}$. Since

![](https://cdn.mathpix.com/cropped/2024_06_05_13ae78702b3aed5f03b4g-7.jpg?height=510&width=772&top_left_y=591&top_left_x=479)

substituting into the second equation of (1), we get

$$
\sqrt{(u+2)^{2}+v^{2}}=2+\sqrt{(u-2)^{2}+v^{2}}
$$

i.e., $u^{2}-\frac{v^{2}}{3}=1$.

Therefore, the required geometric locus of points is a hyperbola.