[ Second-order curves

Through each point $X$, lying inside a given circle $S$, a line $l$ is drawn orthogonal to the line $X O$, where $O$ is a given point not lying on the circle $S$. Describe the set swept out by all lines $l$.

#
We can assume that the center of the circle $S$ is located at the origin, and the point $X$ has coordinates $(c, 0)$. The point $A=(x, y)$ belongs to the desired set if and only if the circle $S$ intersects the circle $S_{1}$ with diameter $AO$. Let $a$ be the radius of the circle $S$, $R$ be the radius of the circle $S_{1}$, and $d$ be the distance between the centers of these circles. The circles $S$ and $S_{1}$ intersect if and only if a triangle can be formed from the segments $a$, $d$, and $R$, i.e.,

$$
(R-a)^{2} \leqslant d^{2} \leqslant(R+a)^{2}
$$

Considering that $4 d^{2}=(x+c)^{2}+y^{2}$ and $4 R^{2}=(x-c)^{2}+y^{2}$, we arrive at the inequality

$$
a^{2}-2 R a \leqslant c x \leqslant 2 R a+a^{2}
$$

which is equivalent to the inequality $\left(c x-a^{2}\right)^{2} \leqslant 4 a^{2} R^{2}$, i.e.,

$$
\left(c^{2}-a^{2}\right) x^{2}-a^{2} y^{2} \leqslant a^{2}\left(c^{2}-a^{2}\right)
$$