Question 4
Let $ABCD$ be a square of side $2a$. In the plane through $AB$ perpendicular to the square's plane, construct an equilateral triangle $AMB$. A point $S$ moves along $AB$ with $SB=x$. Let $P$ be the orthogonal projection of $M$ onto the line $SC$, and let $E,O$ be the midpoints of $AB,CM$, respectively.
(a) Determine the locus of $P$ as $S$ moves along $AB$.
(b) Find the maximum and minimum values of $SO$.
(a) Determine the locus of $P$ as $S$ moves along $AB$.
(b) Find the maximum and minimum values of $SO$.