Curve Traced by a Sliding Linkage

A drawing tool is built as follows. $O$ is the midpoint of a straight slot $AB$. A short rod $ON$ can rotate about $O$, and a long rod $MN$ is joined to $ON$ by a hinge at $N$; a pin $D$ on the rod $MN$ can slide along the slot $AB$. The lengths satisfy $DN=ON=1$ and $MN=3$. As the pin $D$ moves back and forth inside the slot $AB$, the point $N$ makes one full revolution about $O$ (when $D$ is stationary, $N$ is also stationary). The tip $M$ of the long rod traces a curve $C$. Take $O$ as the origin and the line $AB$ as the $x$-axis.

1. Find the equation of the curve $C$.

2. A variable line $l$ meets the two fixed lines $l_1:\; x-2y=0$ and $l_2:\; x+2y=0$ at points $P$ and $Q$ respectively. If $l$ always has exactly one common point with the curve $C$, determine whether the area of $\triangle OPQ$ attains a minimum value. If it does, find that minimum; if not, explain why.