A) The line of intersection of \[{{P}_{1}}\,\,and\,\,{{P}_{2}}\] has direction ratios 1, 2, -1.
B) The line \[\frac{3x-4}{9}=\frac{1-3y}{9}=\frac{z}{3}\] is perpendicular to the line of intersection of \[{{\operatorname{P}}_{1}}\,\,and\,\,{{P}_{2}}\].
C) The acute angle between \[{{\operatorname{P}}_{1}}\,\,and\,\,{{P}_{2}}\] is \[45{}^\circ \].
D) If \[{{P}_{3}}\] is the plane passing through the point (4, 2, -2) and perpendicular to the line of intersection of \[{{\operatorname{P}}_{1}}\,\,and\,\,{{P}_{2}}\], then the distance of the point (2, 1, 1) from the plane \[{{P}_{3}}\,\,is\,\,\frac{2}{\sqrt{3}}\].
Correct Answer: D
Solution :
[a] D.C. of line of intersection (a, b, c) \[\Rightarrow \,\,\,2a+b-c=0\,\,and\,\,a+2b+c=0\] \[\frac{a}{1+2}=\frac{b}{-1-2}=\frac{c}{4-1}\] \[\therefore \] D.C. is (1, -1, 1) [b] \[\frac{3x-4}{9}=\frac{1-3y}{9}=\frac{z}{3}\] \[\Rightarrow \,\,\frac{x-4/3}{3}=\frac{y-1/3}{-3}=\frac{z}{3}\] \[\Rightarrow \] lines are parallel [c] Acute angle between \[{{P}_{1}}\] and \[{{P}_{2}}={{\cos }^{-1}}\left( \frac{2\times 1+1\times 2-1\times 1}{\sqrt{6}\sqrt{6}} \right)\] \[={{\cos }^{-1}}\left( \frac{3}{6} \right)={{\cos }^{-1}}\left( \frac{1}{2} \right)=60{}^\circ \] [d] Plane is given by \[\left( x-4 \right)-\left( y\,-2 \right)+\left( z+2 \right)=0\] \[\Rightarrow \,\,\,x-y+z=0\] Distance of (2, 1, 1) from plane \[=\,\,\frac{2-1+1}{\sqrt{3}}=\frac{2}{\sqrt{3}}\]
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