Key Concepts and Formulae

This problem involves finding the center of a circle based on two conditions: passing through a given point and touching a parabola at another given point. The core mathematical concepts and formulas involved are:

1. Tangent to a Curve: The slope of the tangent line to a curve $y = f(x)$ at a point $(x_1, y_1)$ is given by the derivative $\frac{dy}{dx}$ evaluated at that point, i.e., $m = \left. \frac{dy}{dx} \right|_{(x_1, y_1)}$.
2. Geometric Property of Tangent and Radius: When a circle touches another curve (or a line) at a point, the radius of the circle drawn