Trong không gian với hệ tọa độ Oxyz, cho hai đường thẳng \({{d}_{1}}:\frac{x-1}{2}=\frac{y}{1}=\frac{z+2}{-1}\) và \({{d}_{2}}:\frac{x-1}{1}=\frac{y+2}{3}=\frac{z-2}{-2}\). Gọi \(\...

* Hướng dẫn giải

\(A\in {{d}_{1}}\Rightarrow A\left( 1+2a;a;-2-a \right),B\in {{d}_{2}}\Rightarrow B\left( 1+b;-2+3b;2-2b \right)\).

\(\overrightarrow{AB}\left( b-2a;3b-a-2;-2b+a+4 \right)\).

(P) có vtpt \(\overrightarrow{n}\left( 1;1;1 \right)\).

\(\begin{align} & \Delta //\left( P \right)\Rightarrow \overrightarrow{AB}.\overrightarrow{n}=0\Leftrightarrow b=a-2\Rightarrow \overrightarrow{AB}\left( -a-1;2a-5;-a+6 \right) \\ & \Rightarrow A{{B}^{2}}=6{{a}^{2}}-30a+62\ge 6{{\left( a-\frac{5}{2} \right)}^{2}}+\frac{49}{2}\ge \frac{49}{2} \\ \end{align}\)

\(A{{B}_{\min }}\) khi \(a = \frac{5}{2} \Rightarrow A\left( {6;\frac{5}{2};\frac{{ - 9}}{2}} \right),\overrightarrow {AB} = \frac{7}{2}\left( { - 1;0;1} \right) \Rightarrow \Delta :\left\{ \begin{array}{l} x = 6 - t\\ y = \frac{5}{2}\\ z = \frac{{ - 9}}{2} + t. \end{array} \right.\)