Question

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{1}=\frac{z}{1}$ and intersect, then k =
[IIT Screening 2004]

• A. $\frac{2}{9}$
• B. $\frac{2}{9}$
• C. $0$
• D. $Noneofthese$

Answer 1

The correct option is B

$\frac{2}{9}$

Any point on $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}=\mu$ is, $\mu +3,2\mu +k,\mu );\mu ϵR$
The given lines intersect if and only if the system of equations (in $\lambda$ and $\mu$)
2$\lambda$+1=$\mu$+3 ....(i)
3$\lambda$-1=2$\mu$+k ....(ii)
4$\lambda$+1=$\mu$ .....(iii)
has a unique solution.
Solving (i) and (iii), we get $\lambda =\frac{-3}{2},\mu =-5$
From (ii), we get $\frac{-9}{2}-1=-10+k\Rightarrow k=\frac{9}{2}$