Trong không gian Oxyz, cho hai điểm \(A\left( { - 1;3; - 2} \right);\,\,B\left( { - 3;7; - 18} \right)\) và mặt phẳng \(\left( P \right):\,\,2x - y + z + 1 = 0\). Điểm \(M\left( {a;b;c} \right)\) thuộc (P) sao cho mặt phẳng (ABM) vuông góc với (P) và \(M{A^2} + M{B^2} = 246\). Tính \(S = a + b + c\).

\(M \in \left( P \right) \Rightarrow 2a - b + c + 1 = 0\)

Ta có:

\(\begin{array}{l}\overrightarrow {AB}  = \left( { - 2;4; - 16} \right);\,\,\overrightarrow {AM}  = \left( {a + 1;b - 3;c + 2} \right) \Rightarrow \left[ {\overrightarrow {AB} ;\overrightarrow {AM} } \right] = \left( {16b + 4c - 40;\,\, - 16a + 2c - 12;\,\, - 4a - 2b + 2} \right)\\{\overrightarrow n _{\left( P \right)}} = \left( {2; - 1;1} \right)\\ \Rightarrow 2\left( {16b + 4c - 40} \right) - \left( { - 16a + 2c - 12} \right) + \left( { - 4a - 2b + 2} \right) = 0\\ \Leftrightarrow 12a + 30b + 6c = 66 \Leftrightarrow 2a + 5b + c = 11\\M{A^2} + M{B^2} = 246\\ \Leftrightarrow {\left( {a + 1} \right)^2} + {\left( {b - 3} \right)^2} + {\left( {c + 2} \right)^2} + {\left( {a + 3} \right)^2} + {\left( {b - 7} \right)^2} + {\left( {c + 18} \right)^2} = 246\\ \Leftrightarrow {a^2} + {b^2} + {c^2} + 4a - 10b + 20c + 75 = 0\end{array}\)

Khi đó ta có hệ phương trình \(\left\{ \begin{array}{l}2a - b + c =  - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right)\\2a + 5b + c = 11\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 2 \right)\\{a^2} + {b^2} + {c^2} + 4a - 10b + 20c + 75 = 0\,\,\,\,\,\,\,\,\,\,\,\,\left( 3 \right)\end{array} \right.\)

\(\left( 1 \right),\left( 2 \right) \Rightarrow b = 2 \Rightarrow 2a - 2 + c =  - 1 \Leftrightarrow 2a + c = 1 \Leftrightarrow c = 1 - 2a\)

Thay vào (3) ta có

\({a^2} + 4 + {\left( {1 - 2a} \right)^2} + 4a - 10.2 + 20\left( {1 - 2a} \right) + 75 = 0 \Leftrightarrow 5{a^2} - 40a + 80 = 0 \Leftrightarrow {a^2} - 8a + 16 = 0 \Leftrightarrow a = 4 \Rightarrow c =  - 7\)

Vậy \(S = a + b + c = 4 + 2 - 7 =  - 1\).