Đáp án:
\(C\left( { - \dfrac{3}{2};\dfrac{9}{4}} \right)\).
Giải thích các bước giải:
Gọi \(C\left( {x;y} \right)\).
\(O\) là trực tâm tam giác \(ABC \Rightarrow \left\{ \begin{array}{l}OA \bot BC\\OB \bot AC\end{array} \right. \Rightarrow \left\{ \begin{array}{l}\overrightarrow {OA} .\overrightarrow {BC} = 0\\\overrightarrow {OB} .\overrightarrow {AC} = 0\end{array} \right.\) (1).
Ta có:
\(\begin{array}{l}\overrightarrow {OA} = \left( { - 1;2} \right);\,\,\overrightarrow {BC} = \left( {x - 2;y - 4} \right)\\\overrightarrow {OB} = \left( {2;4} \right);\,\,\overrightarrow {AC} = \left( {x + 1;y - 2} \right)\end{array}\).
\(\begin{array}{l}\left( 1 \right) \Leftrightarrow \left\{ \begin{array}{l} - 1\left( {x - 2} \right) + 2\left( {y - 4} \right) = 0\\2\left( {x + 1} \right) + 4\left( {y - 2} \right) = 0\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l} - x + 2 + 2y - 8 = 0\\2x + 2 + 4y - 8 = 0\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l} - x + 2y = 6\\2x + 4y = 6\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = - \dfrac{3}{2}\\y = \dfrac{9}{4}\end{array} \right.\end{array}\)
Vậy \(C\left( { - \dfrac{3}{2};\dfrac{9}{4}} \right)\).
