Question

[Nguyễn Ngọc Trâm]

trong mp Oxy :A(-1;-4) B(3;4) C(2;5)

a) tìm P khác B sao cho A P B thẳng hàng và AP = \(3\sqrt{5}\)

b) tìm M thuộc Ox sao cho (MA + MB)min

c) tìm M thuộc Ox sao cho (MB+MC) min

Answer 1

[Nguyễn Việt Lâm]

a/ \(\overrightarrow{AB}=\left(4;8\right)\Rightarrow\) đường thẳng AB có 1 vtpt là \(\left(2;-1\right)\)

Phương trình AB:

\(2\left(x-3\right)-\left(y-4\right)=0\Leftrightarrow2x-y-2=0\)

A;P;B thẳng hàng \(\Rightarrow P\in AB\Rightarrow P\left(x;2x-2\right)\)

\(\overrightarrow{AP}=\left(x+1;2x+2\right)\Rightarrow AP^2=\left(x+1\right)^2+\left(2x+2\right)^2=5\left(x+1\right)^2\)

\(\Rightarrow5\left(x+1\right)^2=\left(3\sqrt{5}\right)^2\Rightarrow\left(x+1\right)^2=9\Rightarrow\left[{}\begin{matrix}x=2\\x=-4\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}P\left(2;2\right)\\P\left(-4;-10\right)\end{matrix}\right.\)

Answer 2

[Nguyễn Việt Lâm]

Gọi \(M\left(x;0\right)\)

b/ \(\overrightarrow{AM}=\left(x+1;4\right)\Rightarrow MA=\sqrt{\left(x+1\right)^2+4^2}\)

\(\overrightarrow{MB}=\left(3-x;4\right)\Rightarrow MB=\sqrt{\left(3-x\right)^2+4^2}\)

\(T=MA+MB=\sqrt{\left(x+1\right)^2+4^2}+\sqrt{\left(3-x\right)^2+4^2}\)

Áp dụng BĐT Mincopxki:

\(T\ge\sqrt{\left(x+1+3-x\right)^2+\left(4+4\right)^2}=4\sqrt{5}\)

\(T_{min}=4\sqrt{5}\) khi \(x+1=3-x\Rightarrow x=1\Rightarrow M\left(1;0\right)\)

c/ Tương tự như câu b:

\(MB+MC=\sqrt{\left(3-x\right)^2+4^2}+\sqrt{\left(x-2\right)^2+5^2}\)

\(MB+MC\ge\sqrt{\left(3-x+x-2\right)^2+\left(4+5\right)^2}=\sqrt{82}\)

Dấu "=" xảy ra khi \(\frac{3-x}{4}=\frac{x-2}{5}\Rightarrow x=\frac{23}{9}\Rightarrow M\left(\frac{23}{9};0\right)\)