Which of the following statements are correct?
(A) If \(\displaystyle\frac{1}{c}, \frac{1}{c}, \frac{1}{c}\) are the direction cosines of a line, then \(c = \pm\sqrt{3}\)
(B) The foot of the perpendicular drawn from the point (3, 1, 2) to the x-axis is (3, 0, 0)
(C) The angle between the lines \(\displaystyle\frac{x-1}{1}=\frac{y+2}{1}=\frac{z}{2}\) and \(\displaystyle\frac{x+3}{2}=\frac{y-1}{-1}=\frac{z-4}{1}\) is \(\displaystyle\frac{\pi}{6}\)
(D) If a line makes angles \(\alpha, \beta, \gamma\) with the positive directions of the x-axis, y-axis and z-axis respectively, then \(\sin^2\alpha+\sin^2\beta+\sin^2\gamma = 1\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
For direction cosines l, m, n, the basic relation is \(l^2+m^2+n^2 = 1\). With \(l = m = n = \frac{1}{c}\):
\( \displaystyle \frac{3}{c^2} = 1 \)
\(c^2 = 3\)
A point of the x-axis is (x, 0, 0). The foot of the perpendicular from (3, 1, 2) is the point where the joining vector is perpendicular to the x-axis, direction (1, 0, 0):
\((3-x)(1)+(1)(0)+(2)(0) = 0\)
\(x = 3\)
The lines in (C) have direction ratios 1, 1, 2 and 2, −1, 1:
\( \displaystyle \cos\theta = \frac{(1)(2)+(1)(-1)+(2)(1)}{\sqrt{6}\,\sqrt{6}} = \frac{3}{6} = \frac{1}{2} \)
\( \displaystyle \theta = \frac{\pi}{3} \)
For (D), using \(\sin^2 = 1-\cos^2\) and \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1\):
\(\sin^2\alpha+\sin^2\beta+\sin^2\gamma = 3-1 = 2\)
So \(c^2 = 3\), the foot is (3, 0, 0), the angle between the lines is \(\frac{\pi}{3}\) and \(\sin^2\alpha+\sin^2\beta+\sin^2\gamma = 2\).
Checking the statements
(A) If \(\frac{1}{c}, \frac{1}{c}, \frac{1}{c}\) are the direction cosines of a line, then \(c = \pm\sqrt{3}\) — Correct. \(c^2 = 3\) gives \(c = \pm\sqrt{3}\).
(B) The foot of the perpendicular drawn from the point (3, 1, 2) to the x-axis is (3, 0, 0) — Correct. The foot has x = 3 and y = z = 0.
(C) The angle between the two given lines is \(\frac{\pi}{6}\) — Incorrect. \(\cos\theta = \frac{1}{2}\), so the angle is \(\frac{\pi}{3}\).
(D) \(\sin^2\alpha+\sin^2\beta+\sin^2\gamma = 1\) — Incorrect. The sum is \(3-1 = 2\).