This problem combines concepts from parabolas and circles, specifically the properties of a focal chord and the condition for a line to be tangent to a circle. We will systematically break down the problem to ensure clarity and accuracy.

1. Introduction to Key Concepts and Strategy

To solve this problem, we need to understand and apply the following key concepts:

• Parabola Properties: How to find the focus of a given parabola.
• Focal Chord Definition: A line segment that passes through the focus of a parabola and has its endpoints on the parabola. In this problem, the given line $y=mx+c$ is a focal chord, meaning it must pass through the focus.
• Circle Properties: How to identify the center and radius from the equation of