Consider the line \(L: \displaystyle\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{2}\).
Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Condition | Value of p | ||
| (A) | L is parallel to a line with direction ratios 4, p, 4 | (I) | 4 |
| (B) | L is perpendicular to a line with direction ratios p, 4, 1 | (II) | −1 |
| (C) | L is perpendicular to a line with direction ratios 3, p, −1 | (III) | −2 |
| (D) | \(\displaystyle\frac{2}{3}, \frac{p}{3}, \frac{2}{3}\) are direction cosines of L | (IV) | 1 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
The direction ratios of L are 2, −1, 2. Lines are parallel when the direction ratios are proportional and perpendicular when \(a_1a_2+b_1b_2+c_1c_2 = 0\).
(A) Parallel to 4, p, 4
This matches (III) −2:
\( \displaystyle \frac{4}{2} = \frac{p}{-1} = \frac{4}{2} \)
\(p = -2\)
(B) Perpendicular to p, 4, 1
This matches (IV) 1:
\(2p-4+2 = 0\)
\(p = 1\)
(C) Perpendicular to 3, p, −1
This matches (I) 4:
\(6-p-2 = 0\)
\(p = 4\)
(D) \(\frac{2}{3}, \frac{p}{3}, \frac{2}{3}\) are direction cosines of L
This matches (II) −1. Dividing 2, −1, 2 by \(\sqrt{4+1+4} = 3\):
\( \displaystyle \frac{2}{3}, -\frac{1}{3}, \frac{2}{3} \)
The other set of direction cosines starts with \(-\frac{2}{3}\), so with first value \(\frac{2}{3}\) the middle one must be \(-\frac{1}{3}\), giving \(p = -1\).