Gọi \(I\left( {a;b;c} \right)\) là tâm mặt cầu tiếp xúc với \(\left( {Oyz} \right)\) đồng thời đi qua \(M,N,P\).

Ta có : \(\left\{ \begin{array}{l}IM = IN\\IM = IP\\d\left( {I;\left( {Oyz} \right)} \right) = IN\end{array} \right.\).

Ta có: 

\(\begin{array}{l}\overrightarrow {IM}  = \left( {2 - a;1 - b;4 - c} \right)\\\overrightarrow {IN}  = \left( {5 - a; - 3 - b;1 - c} \right)\\\overrightarrow {IP}  = \left( {1 - a; - 3 - b;1 - c} \right)\\d\left( {I;\left( {Oyz} \right)} \right) = \left| a \right|\end{array}\)

\(\begin{array}{l} \Rightarrow \left\{ \begin{array}{l}{\left( {2 - a} \right)^2} + \left( {1 - {b^2}} \right) + {\left( {4 - c} \right)^2} = {\left( {5 - a} \right)^2} + {b^2} + {c^2}\\{\left( {2 - a} \right)^2} + \left( {1 - {b^2}} \right) + {\left( {4 - c} \right)^2} = {\left( {1 - a} \right)^2} + {\left( {3 + b} \right)^2} + {\left( {1 - c} \right)^2}\\{a^2} = {\left( {5 - a} \right)^2} + {b^2} + {c^2}\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l} - 4a + 4 - 2b + 1 - 8c + 16 =  - 10a + 25\\ - 4a + 4 - 2b + 1 - 8c + 16 =  - 2a + 1 + 6b + 9 - 2c + 1\\{a^2} = {\left( {5 - a} \right)^2} + {b^2} + {c^2}\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}6a - 2b - 8c = 4\\ + 2a + 8b + 6c = 10\\ - 10a + {b^2} + {c^2} =  - 25\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}b = 1 - c\\a = 1 + c\\ - 10\left( {1 + c} \right) + {\left( {1 - c} \right)^2} + {c^2} =  - 25\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}b = 1 - c\\a = 1 + c\\2{x^2} - 12c + 16 = 0\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}\left\{ \begin{array}{l}c = 2\\a = 3\\b =  - 1\end{array} \right.\,\,\left( {tm} \right)\\\left\{ \begin{array}{l}c = 4\\a = 5\\b =  - 3\end{array} \right.\,\,\left( {ktm} \right)\end{array} \right. \Rightarrow c = 2\end{array}\)

Chọn B.