Step-by-Step Solution
Key Concept: A common tangent to both a parabola and circle must satisfy the tangent equations of both curves simultaneously.
The tangent to parabola $y = mx + \frac{1}{m}$. The tangent to circle $y = m \pm \frac{1}{2}\sqrt{1+m^2}$. For common tangent, $\frac{1}{m} = 1(1 + m^2) - 2 = 0$, which gives $m^2 - 1 \Rightarrow m = 1$. Substituting $m=1$, the common tangent is $y = x + 1$.
Correct Answer: Option A