Consider the lines \(L_1: \displaystyle\frac{x}{1}=\frac{y}{1}=\frac{z}{1}\) and \(L_2: \displaystyle\frac{x}{1}=\frac{y}{-1}=\frac{z}{1}\).
(A) \(L_1\) and \(L_2\) intersect at the origin
(B) The acute angle between \(L_1\) and \(L_2\) is \(\cos^{-1}\left(\frac{1}{3}\right)\)
(C) The point (2, −2, 2) lies on \(L_2\)
(D) The shortest distance between \(L_1\) and \(L_2\) is \(\sqrt{2}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (C) only
Both equations are satisfied by x = y = z = 0, so both lines pass through the origin. Their direction ratios are 1, 1, 1 and 1, −1, 1:
\( \displaystyle \cos\theta = \frac{|1-1+1|}{\sqrt{3}\,\sqrt{3}} = \frac{1}{3} \)
For (2, −2, 2) in \(L_2\):
\( \displaystyle \frac{2}{1} = \frac{-2}{-1} = \frac{2}{1} = 2 \)
Lines that intersect have shortest distance 0.
So the lines meet at the origin, \(\cos\theta = \frac{1}{3}\), (2, −2, 2) lies on \(L_2\), and the shortest distance is 0.
Checking the statements
(A) \(L_1\) and \(L_2\) intersect at the origin — Correct. (0, 0, 0) satisfies both equations.
(B) The acute angle between \(L_1\) and \(L_2\) is \(\cos^{-1}\left(\frac{1}{3}\right)\) — Correct. \(\cos\theta = \frac{1}{3}\).
(C) The point (2, −2, 2) lies on \(L_2\) — Correct. All three fractions equal 2.
(D) The shortest distance between \(L_1\) and \(L_2\) is \(\sqrt{2}\) — Incorrect. The lines intersect, so the distance is 0.