Question

Cho A(1,2)B(-3,5) (∆) {x=3t ; y = 2 -t Tìm M thuộc (∆) sao cho MA^2 + MB^2 bé nhất

Answer 1

Do M thuộc \(\Delta\) nên tọa độ có dạng \(M\left(3t;2-t\right)\)

\(\Rightarrow\left\{{}\begin{matrix}\overrightarrow{AM}=\left(3t-1;-t\right)\\\overrightarrow{BM}=\left(3t+3;-t-3\right)\end{matrix}\right.\)

Đặt \(P=MA^2+MB^2=\left(3t-1\right)^2+\left(-t\right)^2+\left(3t+3\right)^2+\left(-t-3\right)^2\)

\(P=20t^2+18x+19=20\left(t+\dfrac{9}{20}\right)^2+\dfrac{299}{20}\ge\dfrac{299}{20}\)

Dấu = xảy ra khi \(t=-\dfrac{9}{20}\Rightarrow M\left(-\dfrac{27}{20};\dfrac{49}{20}\right)\)