Question

If a plane $\pi$ passes through the point $(1,2,3)$ direction cosines of the normal to $\pi$ are $l$, $m$, $n$; and it contains the line joining the origin to the point $(1,1,1)$, then

• A. $l+2m+3n=0$
• B. $l+m+n=0$
• C. $l+m-n=0$
• D. $l-m+2n=0$

Answer 1

Answer: (A), (B)

The correct options are
A $l+2m+3n=0$
B $l+m+n=0$

Given that plane $\pi$ passes through the point $(1,2,3)$, direction cosines of the normal to $\pi$ are $l$, $m$, $n$
Equation of plane is $lx+my+nz=l+2m+3n$ -----$\left(1\right)$
But, above plane contains the line joining origin and $(1,1,1)$.
$\Rightarrow$ $(0,0,0)$ and $(1,1,1)$ lies on $\left(1\right)$
$\Rightarrow l+2m+3n=0$ and $l+m+n=l+2m+3n$
$\therefore l+2m+3n=0$ and $l+m+n=0$
Hence, option A and B.