The direction cosines l, m, n of a line satisfy \(l : m = 1 : 2\), and the line is perpendicular to the line whose direction ratios are 2, 1, 2. The direction cosines of the line can be
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle \frac{1}{3}, \frac{2}{3}, -\frac{2}{3}\)
Since \(l : m = 1 : 2\), let \(l = t\) and \(m = 2t\). The perpendicular condition with 2, 1, 2 gives
\(2t+2t+2n = 0\)
\(n = -2t\)
So the direction ratios are 1, 2, −2. Dividing by \(\sqrt{1+4+4} = 3\):
\( \displaystyle \pm\left(\frac{1}{3}, \frac{2}{3}, -\frac{2}{3}\right) \)
Why the other options are incorrect
Option (1) \(\displaystyle \frac{1}{3}, \frac{2}{3}, \frac{2}{3}\) — Incorrect. \(2\cdot\frac{1}{3}+\frac{2}{3}+2\cdot\frac{2}{3} = \frac{8}{3} \ne 0\), so it is not perpendicular to 2, 1, 2.
Option (2) \(\displaystyle \frac{2}{3}, \frac{1}{3}, -\frac{2}{3}\) — Incorrect. Here \(l : m = 2 : 1\), not \(1 : 2\).
Option (3) \(\displaystyle \frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}}, 0\) — Incorrect. \(\frac{2}{\sqrt{5}}+\frac{2}{\sqrt{5}}+0 \ne 0\), so it is not perpendicular to 2, 1, 2.