This problem asks us to find a relationship between the slope ($m$) and y-intercept ($c$) of a line that is a common tangent to a given hyperbola and a given circle. The core idea is to apply the specific conditions for a line to be tangent to each conic section and then equate the expressions for $c^2$.

1. Key Concepts: Conditions for Tangency

A line $y = mx + c$ is tangent to a conic section if it satisfies a specific condition related to its slope ($m$), y-intercept ($c$), and the parameters of the conic.

• For a Hyperbola: The general equation of a hyperbola centered at the origin is $\frac{x^2}{a^2} - \frac{y