The lines \(\displaystyle\frac{x-1}{1}=\frac{y+1}{2}=\frac{z-2}{-1}\) and \(\displaystyle\frac{x-1}{1}=\frac{y-k}{-1}=\frac{z+1}{2}\) intersect. The value of k is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2
A general point of each line is
\(P(1+s,\ -1+2s,\ 2-s)\)
\(Q(1+t,\ k-t,\ -1+2t)\)
At the point of intersection P and Q are the same point. Comparing x and z:
\(1+s = 1+t\) gives \(s = t\)
\(2-s = -1+2s\)
\(3s = 3\)
\(s = 1,\quad t = 1\)
The y-coordinates must also agree:
\(-1+2(1) = k-1\)
\(k = 2\)
The lines then meet at (2, 1, 1). Their direction ratios 1, 2, −1 and 1, −1, 2 are not proportional, so for any other value of k the lines neither meet nor are parallel.