This problem requires a strong understanding of geometric properties, especially tangency conditions for circles with lines and parabolas, combined with an appreciation for symmetry to simplify the problem. The goal is to find the circle of minimum area, which implies finding the circle with the minimum radius.

1. Understanding the Geometry and Initial Symmetry Analysis

The problem involves a parabola and two lines. Let's analyze their properties:

• Parabola: $y = 6 - x^2$. This is a downward-opening parabola with its vertex at $(0, 6)$. It is symmetric about the y-axis.
• Lines: $y = \sqrt{3}|x|$ represents two lines:
  • $y = \sqrt{3}x$