# The value of $k$ for which the line $\overrightarrow r=(4\hat i+2\hat j-\hat k)+\lambda(\hat i-k\hat j+2\hat k)$ lies on the plane $2x-4y+z=3$ is ?

$(a)\;1\qquad(b)\;-1\qquad(c)\;No\;real\;value\;of\;k\qquad(d)\;3$

For any line to lie on a plane, two conditions are to be satisfied.
1. Every point on the line ($\overrightarrow a$) should satisfy the eqn. of the plane.
2. The line $(\overrightarrow b)$ should be $\perp$ to the normal ($\overrightarrow n$) to the plane.
$i.e.,\:\:\overrightarrow n.\overrightarrow b=0$
Here $\overrightarrow a=(4,2,-1),\:\:\overrightarrow b=(1,k,2)$ and $\overrightarrow n=(2,-4,1)$
The point on the line $(4,2,-1)$ does not satisfy the eqn. of the plane.
$\therefore$ The line does not lie on the plane for any real value of $k$.
answered Jan 6, 2014