Three mutually perpendicular· lines are drawn from the point $(1, 2,-1 ).$ If one of the lines is perpendicular to the $x$-axis and the direction ratios of the second line are $(1, 2,-1),$ then which of the following is/ are possibly the equation(s) of the third line?  

Let the direction ratios of the first and third 

lines be $\left(0,m_1,n_1\right)\text{ and }\left(l_2,m_2,n_2\right)$

Direction ratios of second line are 1, 2 -1. 

Since lines are  perpendicular, 

$2m_1-n_1=0,l_2+2m_2-n_2=0$

On solving, we get 

$\Rightarrow \frac{l_1}{0}=\frac{m_1}{1}=\frac{n_1}{2}\text{ and }\frac{l_2}{5}=\frac{m_2}{-2}=\frac{n_2}{1}$

Hence, equation of the required line are given by 

$\frac{x-1}{5}=\frac{y-2}{-2}=\frac{z+1}{1}=\lambda$

For $\lambda =1\text{ and }\lambda =-1\text{, we get (a) and (c) }$