[ [ curves of the second order ]

Given a point $O$ and a line $l$. A point $X$ moves along the line $l$. Describe the set swept by the perpendiculars to the line $X O$, erected from the point $X$.

#
Let's choose a coordinate system such that the line $l$ is given by the equation $x=0$, and the point $O$ has coordinates (1, 0). A point $X=(x, y)$ belongs to the desired set if and only if the circle with diameter $O X$ intersects the line $l$. This means that the distance from the center of this circle to the line $l$ does not exceed its radius, i.e.,

$$
\left(\frac{x+1}{2}\right)^{2} \leqslant\left(\frac{x-1}{2}\right)^{2}+\frac{y^{2}}{4}
$$

Thus, the desired set is given by the inequality $y^{2} \geqslant 4 x$.