Let \( O \) be the origin, and \( P \) and \( Q \) be two points on the rectangular hyperbola \( xy = 12 \) such that the mid-point of the line segment \( PQ \) is \( \left( \frac{1}{2}, -\frac{1}{2} \right) \). Then the area of the triangle \( OPQ \) equals:
