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 are the possible equation(s) of the third line

Options

A. `vecr=6hati+lambda(5hati-2hatj+hatk)`
B. `(x-1)/5=(y-3)/-2=(z+1)/1`
C. `(x+4)/5=(y-4)/-2=(z+2)/1`
D. none of these

Answer

A, C

Explanation

Let the direction ratios of the first and third lines be `(0,m,n_(1))` and `(l_(2),m_(2),n_(2))`.
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`
and `m_(1)m_(2)+n_(1)n_(2)=0`
On solving, we get
`rArr l_(1)/0=m_(1)/1=n_(1)/2` and `l_(5)/5=m_(2)/-2=n_(2)/1`
Hence, equations of the required line are given by
`(x-1)/5=(y-2)/-2=(z+1)/1=lambda`
Hence, equations of the required line are given by
`(x-1)/5 = (y-2)/-2 (z+1)/1=lambda`
For `lambda=1` and `lambda=-1`, we get (a) and (c).