\(\begin{array}{l} Goi\,\,\,P\left( {a;\,\,b} \right)\\ \Rightarrow \left\{ \begin{array}{l} \overrightarrow {PM} = \left( {1 - a;\,\,6 - b} \right)\\ \overrightarrow {PN} = \left( {6 - a;\,\,\,3 - b} \right) \end{array} \right.\\ \Rightarrow \overrightarrow {PM} = 2\overrightarrow {PN} \\ \Leftrightarrow \left( {1 - a;\,\,6 - b} \right) = 2\left( {6 - a;\,\,\,3 - b} \right)\\ \Leftrightarrow \left\{ \begin{array}{l} 1 - a = 2\left( {6 - a} \right)\\ 6 - b = 2\left( {3 - b} \right) \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} 1 - a = 12 - 2a\\ 6 - b = 6 - 2b \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} a = 11\\ b = 0 \end{array} \right.\\ \Rightarrow P\left( {11;\,\,0} \right). \end{array}\)

Đáp án:

Giải thích các bước giải: Ta có : \(\begin{array}{l}P\left( {x;y} \right);\,\,\,\,M\left( {1;6} \right)\,\,\,;\,N\left( {6;3} \right)\\\overrightarrow {PM} = 2\overrightarrow {PN} \\ \Leftrightarrow \left( {1 - x;\,6 - y} \right) = 2.\left( {6 - x;\,3 - y} \right)\\ \Leftrightarrow \left\{ \begin{array}{l}1 - x = 2\left( {6 - x} \right)\\6 - y = 2\left( {3 - y} \right)\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}2x - x = 12 - 1\\2y - y = 6 - 6\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = 11\\y = 0\end{array} \right.\end{array}\) \( \Rightarrow P\left( {11;0} \right)\)