Trong không gian hệ trục tọa độ \(Oxyz\), cho đường thẳng \({\rm{d}}:\frac{{x + 1}}{1} = \frac{{y – 1}}{{ – 1}} = \frac{z}{{ – 2}}\), \(I\left( {{\rm{1;1;1}}} \right)\). Viết phương trình mặt phẳng \(\left( P \right)\) chứa đường thẳng \({\rm{d}}\), đồng thời khoảng cách từ \(I\) đến mặt phẳng \(\left( P \right)\) bằng \(\sqrt 3 \).
A. \(\left( P \right){\rm{:}}\,x – y + z – 2 = 0\), \(\left( P \right){\rm{:}}\,7x + 5y + z + 2 = 0\).
B. \(\left( P \right){\rm{:}}\,x – y + z + 2 = 0\), \(\left( P \right){\rm{:}}\,7x + 5y + z + 2 = 0\).
C. \(\left( P \right){\rm{: }}x – y + z – 2 = 0\), \(\left( P \right){\rm{:}}\,7x + 5y + z – 2 = 0\).
D. \(\left( P \right){\rm{:}}\,x – y + z + 2 = 0\), \(\left( P \right){\rm{:}}\,7x + 5y + z – 2 = 0\).
Lời giải:
Lấy \(M\left( { – 1;1;0} \right)\), \(N\left( {0;0; – 2} \right)\) thuộc đường thẳng \({\rm{d}}\).
Phương trình mặt phẳng \(\left( {\rm{P}} \right)\) có dạng \(ax + by + cz + d = 0,\,\left( {{a^2} + {b^2} + {c^2} \ne 0} \right)\).
Ta có: \(\left\{ \begin{array}{l}M \in \left( P \right)\\N \in \left( P \right)\\{\rm{d}}\left( {I,\left( P \right)} \right) = \sqrt 3 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} – a + b + d = 0\\ – 2c + d = 0\\\frac{{|a + b + c + d|}}{{\sqrt {{a^2} + {b^2} + {c^2}} }} = \sqrt 3 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}d = – a + b\\d = 2c\\\frac{{|a + b + c + d|}}{{\sqrt {{a^2} + {b^2} + {c^2}} }} = \sqrt 3 \end{array} \right.\)
\( \Leftrightarrow \left\{ \begin{array}{l}2c = a – b\\d = a – b\\\left| {a + b + \frac{{a – b}}{2} + a – b} \right| = \sqrt 3 \sqrt {{a^2} + {b^2} + {{\left( {\frac{{a – b}}{2}} \right)}^2}} \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}2c = a – b\\d = a – b\\5{a^2} – 2ab – 7{b^2} = 0\end{array} \right.\)
\( \Leftrightarrow \left\{ \begin{array}{l}2c = a – b\\d = a – b\\\left( {a + b} \right)\left( {5a – 7b} \right) = 0\end{array} \right.\)\( \Leftrightarrow \left[ \begin{array}{l}\left\{ \begin{array}{l}a = – b\\2c = a – b\\d = a – b\end{array} \right.\\\left\{ \begin{array}{l}5a = 7b\\2c = a – b\\d = a – b\end{array} \right.\end{array} \right.\)
Với \(\left\{ \begin{array}{l}a = – b\\2c = a – b\\d = a – b\end{array} \right.\). ộ số \(\left( {a;b;c;d} \right) = \left( {1; – 1;1;2} \right) \Rightarrow \)\(\left( {\rm{P}} \right){\rm{:}}\,x – y + z + 2 = 0\).
Với \(\left\{ \begin{array}{l}5a = 7b\\2c = a – b\\d = a – b\end{array} \right.\). ộ số \(\left( {a;b;c;d} \right) = \left( {7;5;1;2} \right)\)\( \Rightarrow \left( {\rm{P}} \right){\rm{:}}\,7x + 5y + z + 2 = 0\).