Question

[Vương Kỳ Nguyên]

Cho A(-1;0), B(2;3), C(3;-6). Δ: x-2y-3=0. Tìm M thuộc Δ thỏa mãn \(|\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}|\)nhỏ nhất.

Answer 1

[Nguyễn Việt Lâm]

Do \(M\in\Delta\Rightarrow M\left(2m+3;m\right)\)

\(\overrightarrow{MA}=\left(-2m-4;-m\right);\overrightarrow{MB}=\left(-2m-1;3-m\right);\overrightarrow{MC}=\left(-2m;-6-m\right)\)

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

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

\(\Rightarrow P^2=\left(6m+5\right)^2+\left(3m+3\right)^2\)

\(\Rightarrow P^2=36m^2+60m+25+9m^2+18m+9\)

\(\Rightarrow P^2=45m^2+78m+34\)

\(\Rightarrow P^2=45\left(m^2+2.\frac{13}{15}+\frac{169}{225}\right)+\frac{1}{5}\)

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

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

Answer 2

[Vương Kỳ Nguyên]

Kết quả của tôi ra M(-1/3;-5/3)