Question

Cho A(-1;4) , B(1;-2) , C(3;4)

1,Tìm điểm M thuộc Ox sao cho \(\left|\overrightarrow{MA}+2\overrightarrow{MB}\right|\)Nhỏ nhất 2,Tìm P thuộc Oy sao cho \(\left|\overrightarrow{PA}+2\overrightarrow{PB}-4\overrightarrow{PC}\right|\)Nhỏ nhất

Answer 1

Gọi \(M\left(x;0\right)\Rightarrow\left\{{}\begin{matrix}\overrightarrow{MA}=\left(-1-x;4\right)\\\overrightarrow{MB}=\left(1-x;-2\right)\end{matrix}\right.\) \(\Rightarrow\overrightarrow{MA}+2\overrightarrow{MB}=\left(1-3x;0\right)\)

\(\Rightarrow\left|\overrightarrow{MA}+2\overrightarrow{MB}\right|=\sqrt{\left(1-3x\right)^2}\ge0\)

Dấu "=" xảy ra khi \(x=\frac{1}{3}\Rightarrow M\left(\frac{1}{3};0\right)\)

Gọi \(P\left(0;y\right)\) \(\Rightarrow\left\{{}\begin{matrix}\overrightarrow{PA}=\left(-1;4-y\right)\\\overrightarrow{PB}=\left(1;-2-y\right)\\\overrightarrow{PC}=\left(3;4-y\right)\end{matrix}\right.\)

\(\Rightarrow\overrightarrow{PA}+2\overrightarrow{PB}-4\overrightarrow{PC}=\left(-11;5y-16\right)\)

\(\Rightarrow\left|\overrightarrow{PA}+\overrightarrow{PB}-4\overrightarrow{PC}\right|=\sqrt{11^2+\left(5y-16\right)^2}\ge11\)

Dấu "=" xảy ra khi \(5y-16=0\Rightarrow y=\frac{16}{5}\Rightarrow P\left(0;\frac{16}{5}\right)\)