The lines \(\displaystyle\frac{x-1}{1}=\frac{y-2}{-1}=\frac{z-1}{2}\) and \(\displaystyle\frac{x+2}{2}=\frac{y+1}{1}=\frac{z-1}{1}\) intersect. The line through their point of intersection and perpendicular to both of them is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{x-2}{1}=\frac{y-1}{-1}=\frac{z-3}{-1}\)
The general points of the two lines are
\((1+\lambda,\ 2-\lambda,\ 1+2\lambda)\) and \((-2+2\mu,\ -1+\mu,\ 1+\mu)\)
Equating the z-coordinates gives \(\mu = 2\lambda\). Then the x-coordinates give
\(1+\lambda = -2+4\lambda\)
\(\lambda = 1,\quad \mu = 2\)
The y-coordinates agree (\(2-1 = 1\) and \(-1+2 = 1\)), so the lines meet at Q(2, 1, 3).
A direction perpendicular to both lines is the cross product of their directions:
\((\hat{\imath}-\hat{\jmath}+2\hat{k})\times(2\hat{\imath}+\hat{\jmath}+\hat{k}) = (-1-2)\hat{\imath}-(1-4)\hat{\jmath}+(1+2)\hat{k}\)
\(= -3\hat{\imath}+3\hat{\jmath}+3\hat{k}\)
so the direction ratios can be taken as 1, −1, −1. The line through Q(2, 1, 3) is
\( \displaystyle \frac{x-2}{1}=\frac{y-1}{-1}=\frac{z-3}{-1} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{x-1}{1}=\frac{y-2}{-1}=\frac{z-1}{-1}\) — Incorrect. The direction is right, but it passes through (1, 2, 1), the given point of the first line, not through Q(2, 1, 3).
Option (2) \(\displaystyle\frac{x-2}{1}=\frac{y-1}{1}=\frac{z-3}{-1}\) — Incorrect. With the first line, \(1-1-2 = -2 \ne 0\), so it is not perpendicular to it.
Option (3) \(\displaystyle\frac{x+2}{1}=\frac{y+1}{-1}=\frac{z-1}{-1}\) — Incorrect. The direction is right, but it passes through (−2, −1, 1), the given point of the second line, not through Q(2, 1, 3).