If the lines x−12=y+13=z−14 and x−31=y−k1=z1 and intersect, then k =
[IIT Screening 2004]
29
Any point on x−12=y+13=z−14=μ is, μ+3,2μ+k,μ);μϵR
The given lines intersect if and only if the system of equations (in λ and μ)
2λ+1=μ+3 ....(i)
3λ-1=2μ+k ....(ii)
4λ+1=μ .....(iii)
has a unique solution.
Solving (i) and (iii), we get λ=−32,μ=−5
From (ii), we get −92−1=−10+k⇒k=92