Question

Cho 3 điểm A(-6;3) B(0;-1) C(3;2) . M(a;b) là điểm nằm trên d: 2x -y +3=0 sao cho \(|\)\(\overrightarrow{Ma}+\overrightarrow{MB}+\overrightarrow{MC}\)\(_|\) nhỏ nhất.

Answer 1

Do \(M\in d\Rightarrow M\left(a;2a+3\right)\) \(\Rightarrow\left\{{}\begin{matrix}\overrightarrow{MA}=\left(-6-a;-2a\right)\\\overrightarrow{MB}=\left(-a;-4-2a\right)\\\overrightarrow{MC}=\left(3-a;-1-2a\right)\end{matrix}\right.\)

\(\Rightarrow\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}=\left(-3-3a;-5-6a\right)\)

\(\Rightarrow P=\left|\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}\right|=\sqrt{\left(3a+3\right)^2+\left(6a+5\right)^2}\)

\(\Rightarrow P=\sqrt{45a^2+78a+34}=\sqrt{45\left(a+\frac{13}{15}\right)^2+\frac{1}{5}}\ge\sqrt{\frac{1}{5}}\)

\(\Rightarrow P_{min}=\frac{1}{\sqrt{5}}\) khi \(a=-\frac{13}{15}\Rightarrow M\left(-\frac{13}{15};\frac{19}{15}\right)\)