Phương trình mặt cầu (S) đi qua \(A(1;2; – 4),{\rm{ }}B(1; – 3;1),{\rm{ }}C(2;2;3)\) và tâm \(I \in (Oxy)\) là.

Vì \(I \in (Oxy)\) nên gọi I(x;y;0). Ta có: \(\left\{ \begin{array}{l}IA = IB\\IA = IC\end{array} \right.\)

\( \Leftrightarrow \left\{ \begin{array}{l}\sqrt {{{(x – 1)}^2} + {{(y – 2)}^2} + {4^2}} = \sqrt {{{(x – 1)}^2} + {{(y + 3)}^2} + {1^2}} \\\sqrt {{{(x – 1)}^2} + {{(y – 2)}^2} + {4^2}} = \sqrt {{{(x – 2)}^2} + {{(y – 2)}^2} + {3^2}} \end{array} \right.\)

\( \Leftrightarrow \left\{ \begin{array}{l}10y = 10\\2x = – 4\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = – 2\\y = 1\end{array} \right. \Rightarrow I( – 2;1;0) \Rightarrow R = IA = \sqrt {26} \)

\( \Rightarrow {(x + 2)^2} + {(y – 1)^2} + {z^2} = 26\)