Question

A plane $P_1$ has the equation $2 x-y+z=4$ and the plane $P_2$ has the equation $x+n y+2 z=11$ (where $n \in N$ ). If the angle between $P_1$ and $P_2$ is $\frac{\pi}{3}$ then the sum of all possible values of $n$, is

• A. 1
• B. -16
• C. 17
• D. 16

Answer 1

Answer: (A)

${ h _1}=2 \hat{ i }-\hat{ j }+\hat{ k } ; h _2=\hat{ i }+ n \hat{ j }+2 \hat{ k } ; \cos \frac{\pi}{3}=\frac{2- n +2}{\sqrt{6} \sqrt{5+ n ^2}}=\frac{1}{2}$
$n ^2+16 n -17=0 \Rightarrow n =-17$ (reject as $n \in N$ ) or $n =1$