Question

cho các điểm a( -1;2),B(-2;0);C(0,5), điểm M(a,b) thỏa mãn AM+2BM+3CM=0. tính 2a+6b

Answer 1

\(\overrightarrow{AM}=\left(a+1;b-2\right)\)

\(\overrightarrow{BM}=\left(a+2;b\right)\)

\(\overrightarrow{CM}=\left(a;b-5\right)\)

\(\overrightarrow{AM}+2\overrightarrow{BM}+3\overrightarrow{CM}=\overrightarrow{0}\)

\(\Rightarrow\left\{{}\begin{matrix}a+1+2\left(a+2\right)+3a=0\\b-2+2b+3\left(b-5\right)=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}6a=-5\\6b=17\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a=-\dfrac{5}{6}\\b=\dfrac{17}{6}\end{matrix}\right.\)

\(\Rightarrow2a+6b=2.\left(-\dfrac{5}{6}\right)+6.\dfrac{17}{6}=-\dfrac{5}{3}+17=\dfrac{46}{3}\)