Consider the line \(L: \displaystyle\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{6}\). Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Quantity | Value | ||
| (A) | Cosine of the acute angle between L and the z-axis | (I) | \(\displaystyle\frac{4}{21}\) |
| (B) | Value of k for which (5, k, 15) lies on L | (II) | \(\displaystyle\frac{6}{7}\) |
| (C) | Cosine of the acute angle between L and the line with direction ratios 2, −2, 1 | (III) | 8 |
| (D) | Value of p for which L is perpendicular to the line with direction ratios p, 2, −2 | (IV) | 3 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(II), (B)-(III), (C)-(I), (D)-(IV)
L passes through (1, 2, 3) with direction ratios 2, 3, 6, and \(\sqrt{4+9+36} = 7\), so its direction cosines are \(\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\).
(A) Cosine of the acute angle between L and the z-axis
This matches (II) \(\frac{6}{7}\). It is the direction cosine n.
(B) Value of k for which (5, k, 15) lies on L
This matches (III) 8. From x and z:
\( \displaystyle \frac{5-1}{2} = 2,\quad \frac{15-3}{6} = 2 \)
\( \displaystyle \frac{k-2}{3} = 2 \), so \(k = 8\)
(C) Cosine of the acute angle between L and the line with direction ratios 2, −2, 1
This matches (I) \(\frac{4}{21}\):
\( \displaystyle \frac{|4-6+6|}{7\times 3} = \frac{4}{21} \)
(D) Value of p for which L is perpendicular to the line with direction ratios p, 2, −2
This matches (IV) 3:
\(2p+6-12 = 0\), so \(p = 3\)