Đáp án:
a) 2 đường thẳng cắt nhau
Giải thích các bước giải:
\(\begin{array}{l}
vtcp:{\overrightarrow u _{{d_1}}} = \left( {1; - 1} \right) \to vtpt:{\overrightarrow n _{{d_1}}} = \left( {1;1} \right)\\
vtpt:{\overrightarrow n _{{d_2}}} = \left( {2; - 5} \right)\\
a)Xét:\dfrac{1}{2} \ne - \dfrac{1}{5}
\end{array}\)
⇒ 2 đường thẳng cắt nhau
\(\begin{array}{l}
b)Do:\left( \Delta \right)//\left( {{d_2}} \right)\\
\to vtpt:{\overrightarrow n _\Delta } = \left( {2; - 5} \right)\\
PTTQ:\left( \Delta \right):2x - 5y + c = 0\\
Do:d\left( {A;\left( \Delta \right)} \right) = 3\\
\to \dfrac{{\left| {2.3 - 5.5 + c} \right|}}{{\sqrt {{2^2} + {5^2}} }} = 3\\
\to \left| { - 16 + c} \right| = 3\sqrt {29} \\
\to \left[ \begin{array}{l}
- 16 + c = 3\sqrt {29} \\
- 16 + c = - 3\sqrt {29}
\end{array} \right.\\
\to \left[ \begin{array}{l}
c = 3\sqrt {29} + 16\\
c = - 3\sqrt {29} + 16
\end{array} \right.\\
\to \left[ \begin{array}{l}
\left( \Delta \right):2x - 5y + 3\sqrt {29} + 16 = 0\\
\left( \Delta \right):2x - 5y - 3\sqrt {29} + 16 = 0
\end{array} \right.\\
c)\cos \left( {{{\overrightarrow n }_{{d_1}}};{{\overrightarrow n }_{{d_2}}}} \right) = \dfrac{{\left| {{{\overrightarrow n }_{{d_1}}}.{{\overrightarrow n }_{{d_2}}}} \right|}}{{\left| {{{\overrightarrow n }_{{d_1}}}} \right|.\left| {{{\overrightarrow n }_{{d_2}}}} \right|}} = \dfrac{{\left| {1.2 - 1.5} \right|}}{{\sqrt 2 .\sqrt {29} }}\\
= \dfrac{3}{{\sqrt {58} }}\\
\to \left( {{{\overrightarrow n }_{{d_1}}};{{\overrightarrow n }_{{d_2}}}} \right) \approx 66,8^\circ \\
d)Do:\left( a \right) \bot \left( {{d_1}} \right)\\
\to vtpt:{\overrightarrow n _a} = \left( {1; - 1} \right)\\
\to PTTQ:\left( a \right):x - y + c = 0\\
Do:d\left( {B;\left( a \right)} \right) = 2\\
\to \dfrac{{\left| { - 2 - 4 + c} \right|}}{{\sqrt 2 }} = 2\\
\to \left| {c - 6} \right| = 2\sqrt 2 \\
\to \left[ \begin{array}{l}
c - 6 = 2\sqrt 2 \\
c - 6 = - 2\sqrt 2
\end{array} \right.\\
\to \left[ \begin{array}{l}
c = 6 + 2\sqrt 2 \\
c = 6 - 2\sqrt 2
\end{array} \right.\\
\to \left[ \begin{array}{l}
\left( a \right):x - y + 6 + 2\sqrt 2 = 0\\
\left( a \right):x - y + 6 - 2\sqrt 2 = 0
\end{array} \right.
\end{array}\)
