291. Given a right circular cone and a point $A$ outside it, located at a distance from the plane of its base equal to the height of the cone. Let $M$ be a point on the surface of the cone such that a light ray coming from $A$ to $M$, reflecting off the surface of the cone, will be parallel to the plane of the base. Find the geometric locus of the projections of points $M$ on the plane of the base of the cone.
291. Let us introduce the following notations: $\quad S$ - the vertex of the cone, $N$ - the projection of point $M$ onto the plane passing through points $S$ and $A$ parallel to the base of the cone, $P$ - a point on the line $S N$ such that $\widehat{S M P}=90^{\circ}$ (Fig. 56), $MP$ is the normal to the surface of the cone. From the condition, it follows that $A P$ is parallel to the reflected ray. Therefore, $\widehat{A M P}=\widehat{M P A},|A M|=|A P|$. Let $\alpha$ be the angle between the height of the cone and the generatrix, $|S A|=a$. The plane passing through $M$ parallel to the plane $S P A$ intersects the axis of the cone at point $S_{1}, A_{i}$ is the projection of $A$ onto this plane,

$$
\left|S S_{\mathrm{i}}\right|=x, \widehat{M S_{1} A_{i}}=\varphi,\left|M A_{i}\right|=y
$$

By the cosine theorem for $\triangle S_{1} M A_{1}$, we have

$$
y^{2}=x^{2} \operatorname{tg}^{2} \alpha+a^{2}-2 a x \operatorname{tg} \alpha \cos \varphi
$$

Moreover,

$$
\begin{gathered}
|P A|^{2}=|M A|^{2}=y^{2}+x^{2} \\
|S P|=\frac{|S M|}{\sin \alpha}=\frac{x}{\cos \alpha \sin \alpha}=\frac{2 x}{\sin 2 \alpha}
\end{gathered}
$$

Writing the cosine theorem for $\triangle S P A$, using relations (1) and (3), we have

$$
x^{2} \operatorname{tg}^{2} \alpha-2 a x \operatorname{tg} \alpha \cos \varphi+x^{2}=\frac{4 x^{2}}{\sin ^{2} 2 \alpha}-\frac{4 a x}{\sin 2 \alpha} \cos \varphi
$$

from which $x=a \sin 2 \alpha \cos \varphi$.

If we now draw a perpendicular to $S N$ at point $N$ in the plane $S P A$ and denote by $L$ its intersection point with $S A$, then

$$
|S L|=\frac{|S N|}{\cos \varphi}=\frac{x \operatorname{tg} \alpha}{\cos \varphi}=2 a \sin ^{2} \alpha
$$

Thus, $|S L|$ is constant, therefore, point $N$ describes a circle with diameter $S L$.