Question

Cho tam giác ABC có A(1;3); B(-3;1); C(5;-6). Tìm điểm E thuộc Oy sao cho |vecto EA + 3 vecto EB - 2 vecto EC|đạt GTNN

Answer 1

Gọi \(E\left(0;a\right)\Rightarrow\left\{{}\begin{matrix}\overrightarrow{EA}=\left(1;3-a\right)\\\overrightarrow{EB}=\left(-3;1-a\right)\\\overrightarrow{EC}=\left(5;-6-a\right)\end{matrix}\right.\)

\(\Rightarrow T=\overrightarrow{EA}+3\overrightarrow{EB}-2\overrightarrow{EC}=\left(-18;18-2a\right)\)

\(\Rightarrow\left|T\right|=\sqrt{18^2+\left(18-2a\right)^2}\ge18\)

Dấu "=" xảy ra khi \(18-2a=0\Leftrightarrow a=9\)

\(\Rightarrow E\left(0;9\right)\)