question_answer

If the angle\[\theta \]between the line\[\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}\]and  the  plane \[2x-y+\sqrt{\lambda }z+4=0\]is such that \[\sin \theta =\frac{1}{3}\]The value of\[\lambda \]is     AIEEE  Solved  Paper-2005

A) \[-\frac{4}{3}\]                 

B)        \[\frac{3}{4}\]                   

C)        \[-\frac{3}{5}\]                 

D)        \[\frac{5}{3}\]

Correct Answer: D

Solution :

Since, the angle between the line and plane is a complementary angle between the line and normal to the plane. Note The angle between a lineand plane is given by Alternate Solution Direction ratios of line normal are and direction ratios of a plane are Since, \[\sin \theta =\,\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}\,+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}\,\sqrt{a_{2}^{2}\,+b_{2}^{2}\,+c_{2}^{2}}}}\]