Đáp án:

Tìm điểm \( I(x_0, y_0, z_0) \). Khi đó mặt phẳng \( 2x_0 - 3y_0 + z_0 = 0 \) và  

\( d(I, (P)) = \frac{|(1 - m^2)x_0 + 2my_0 + 2(1 + m^2)z_0 + 5m^2 + 4m + 12|}{\sqrt{(1 - m^2)^2 + 4m^2 + 4(1 + m^2)^2}} = R, \quad \forall m \in \mathbb{R}. \)

\( = \frac{|m^2(-x_0 + 2z_0 + 5) + m(2y_0 + 4) + x_0 + 2z_0 + 12|}{\sqrt{5m^4 + 10m^2 + 5}} = R, \quad \forall m \in \mathbb{R}. \)

\( = \frac{|(-x_0 + 2z_0 + 5)m^2 + (2y_0 + 4)m + x_0 + 2z_0 + 12|}{\sqrt{5(m^2 + 1)}} = R, \quad \forall m \in \mathbb{R}. \)

\( \Rightarrow 
\begin{cases}
2y_0 + 4 = 0 \\
-x_0 + 2z_0 + 5 = x_0 + 2z_0 + 12
\end{cases} \Leftrightarrow \begin{cases} y_0 = -2\\ x_0 = -\frac{7}{2} \end{cases} \)

\( \Rightarrow z_0 = -2x_0 + 3y_0 = 7 - 6 = 1 \Rightarrow I\left(-\frac{7}{2}, -2, 1\right). \)

\( \Rightarrow R = \frac{21\sqrt{5}}{10} \Rightarrow \boxed{A} \)