This problem requires us to find a line that is simultaneously tangent to two given curves: a parabola and a circle. Such a line is known as a common tangent. We will leverage the standard forms and tangency conditions for both conic sections to determine its equation.

1. Essential Mathematical Concepts for Solving

Before diving into the solution, let's review the key formulas and concepts we'll be using. A strong understanding of these fundamentals is crucial for solving problems involving tangents to conic sections.

• Equation of a Tangent to a Parabola ($y^2 = 4ax$): For a standard parabola of the form $y^2 = 4ax$, the equation of a tangent line with a slope $m$ is given by: