Question 386 of 521MediumMCQ
The equation of the plane containing the line 2x−5y+z=3, x+y+4z=5, and parallel to the plane x+3y+6z=1, is: Solutions · 1
Official solution
Equation of the plane containing the lines
2x−5y+z=3 and x+y+4z=5 is
\[2x - 5y + z - 3 + \lambda \left( {x + y + 4z - 5} \right) = 0\]
\[ \Rightarrow \left( {2 + \lambda } \right)x + \left( { - 5 + \lambda } \right)y + \left( {1 + 4\lambda } \right)z + \]
\[\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( { - 3 - 5\lambda } \right) = 0....\left( i \right)\]
Since the plane (i) parallel to the given plane
x+3y+6z=1
\[\therefore\] \[\,\,\,{{2 + \lambda } \over 1} = {{ - 5 + \lambda } \over 3} = {{1 + 4\lambda } \over 6}\]
\[\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \lambda = - {{11} \over 2}\]
Hence equation of the required plane is
\[\left( {2 - {{11} \over 2}} \right)x + \left( { - 5 - {{11} \over 2}} \right)y + \left( {1 - {{44} \over 2}} \right)z + \]
\[\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( { - 3 + {{55} \over 2}} \right) = 0\]
\[ \Rightarrow x + 3y + 6z = 7\] Answer the question to unlock