This problem involves a circle tangent to both the x-axis and a parabola. The core idea is to use the geometric properties of tangency to establish relationships between the circle's parameters (center and radius) and the given points/curves.

1. Key Concepts and Formulas

• Equation of a Circle: A circle with center $(h,k)$ and radius $r$ has the equation $(x-h)^2 + (y-k)^2 = r^2$.
• Circle Tangent to x-axis: If a circle touches the x-axis at $(a,0)$, its center must be $(a, \pm r)$, where $r$ is the radius. Since the point of tangency with the parabola $(4,6)$ has a positive