1. Key Concepts and Formulas

To solve this problem, we need to find lines that are simultaneously tangent to a given parabola and a given circle. We will use the following standard formulas and conditions:

• Equation of a Tangent to a Parabola: For a parabola of the form $y^2 = 4Ax$, the general equation of a tangent with slope $m$ is given by $y = mx + \frac{A}{m}$.
• Condition for Tangency to a Circle: A line $Lx + My + N = 0$ is tangent to a circle with center $(h,k)$ and radius $R$ if the perpendicular distance from the center $(h,k)$ to the line is equal to the radius $R$. This is expressed