Question 3
In a plane $P$, $ABCD$ is a square of side $a$. On the ray $Az$ perpendicular to $P$, choose $S$ with $SA=2a$.
(a) Points $M,N$ move on $BC,DC$, respectively.
(i) Find their positions such that $BM+DN\ge3a/2$, the planes $SAM$ and $SMN$ are perpendicular, and $BM\cdot DN$ is least.
(ii) Find their positions such that $\angle NAM=45^\circ$ and the volume of tetrahedron $SAMN$ is greatest or least; calculate these volumes.
(b) A moving point $Q$ sees both $AB$ and $AD$ at right angles. Let $\pi$ be the plane perpendicular to $P$ along $AB$, and let $DQ$ meet $\pi$ at $Q'$.
(i) Find the locus of $Q'$.
(ii) The points $Q$ trace a circle $(K)$. Let $R\ne Q$ be the second intersection of $CQ$ and $(K)$. Prove that $\sin^2\angle Q'DB+\sin^2\angle R'DB$ is constant, where $R'=DR\cap\pi$.
(a) Points $M,N$ move on $BC,DC$, respectively.
(i) Find their positions such that $BM+DN\ge3a/2$, the planes $SAM$ and $SMN$ are perpendicular, and $BM\cdot DN$ is least.
(ii) Find their positions such that $\angle NAM=45^\circ$ and the volume of tetrahedron $SAMN$ is greatest or least; calculate these volumes.
(b) A moving point $Q$ sees both $AB$ and $AD$ at right angles. Let $\pi$ be the plane perpendicular to $P$ along $AB$, and let $DQ$ meet $\pi$ at $Q'$.
(i) Find the locus of $Q'$.
(ii) The points $Q$ trace a circle $(K)$. Let $R\ne Q$ be the second intersection of $CQ$ and $(K)$. Prove that $\sin^2\angle Q'DB+\sin^2\angle R'DB$ is constant, where $R'=DR\cap\pi$.