Question 3
From a point $O$ outside a plane $(P)$, perpendiculars $OH$ are dropped to the lines in $P$ passing through a fixed point $A$. Determine the locus $(c)$ of $H$.
Let $(C)$ be the oblique cone with vertex $O$ and base $(c)$. Prove that planes parallel to $(P)$ or perpendicular to $OA$ cut $(C)$ in circles.
$(C)$ has two planes of symmetry that cut it in angles $\alpha,\beta$. Determine the relation between $\alpha$ and $\beta$.
Editorial conventions: proper circles and a nondegenerate cone require $A$ to differ from the perpendicular projection of $O$ onto $P$, and the section planes must not pass through $O$. “Angles” means full apex angles in the two axial symmetry planes. We state the relation with the smaller one labeled $\alpha$; the source does not specify their order.
Let $(C)$ be the oblique cone with vertex $O$ and base $(c)$. Prove that planes parallel to $(P)$ or perpendicular to $OA$ cut $(C)$ in circles.
$(C)$ has two planes of symmetry that cut it in angles $\alpha,\beta$. Determine the relation between $\alpha$ and $\beta$.
Editorial conventions: proper circles and a nondegenerate cone require $A$ to differ from the perpendicular projection of $O$ onto $P$, and the section planes must not pass through $O$. “Angles” means full apex angles in the two axial symmetry planes. We state the relation with the smaller one labeled $\alpha$; the source does not specify their order.