Đáp án:
\(H\left( {\dfrac{7}{3};\dfrac{{25}}{3}} \right)\).
Giải thích các bước giải:
Gọi \(H\left( {x;y} \right)\) là trực tâm tam giác ABC.
\( \Rightarrow \left\{ \begin{array}{l}AH \bot BC\\BH \bot AC\end{array} \right. \Rightarrow \left\{ \begin{array}{l}\overrightarrow {AH} .\overrightarrow {BC} = 0\\\overrightarrow {BH} .\overrightarrow {AC} = 0\end{array} \right.\) (*)
Ta có:
\(\begin{array}{l}\overrightarrow {AH} = \left( {x - 1;y - 3} \right),\,\,\overrightarrow {BC} = \left( {8; - 2} \right)\\\overrightarrow {BH} = \left( {x + 4;y - 2} \right),\,\,\,\overrightarrow {AC} = \left( {3; - 3} \right)\end{array}\)
Khi đó ta có
\(\begin{array}{l}\left( * \right) \Leftrightarrow \left\{ \begin{array}{l}8\left( {x - 1} \right) - 2\left( {y - 3} \right) = 0\\3\left( {x + 4} \right) - 3\left( {y - 2} \right) = 0\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}8x - 2y - 2 = 0\\3x - 3y + 18 = 0\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}x = \dfrac{7}{3}\\y = \dfrac{{25}}{3}\end{array} \right.\end{array}\)
Vậy \(H\left( {\dfrac{7}{3};\dfrac{{25}}{3}} \right)\).
