Let $P$ be a point on the parabola $y^2 = 12x$ and $N$ be the foot of the perpendicular drawn from $P$, on the axis of the parabola. A line is now drawn through the mid-point $M$ of $PN$, parallel to its axis which meets the parabola at $Q$. If the $y$-intercept of the line $NQ$ is $\frac{1}{2}$, then