Each line in List-I passes through the point P(1, −1, 2). Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Line through P | Equation | ||
| (A) | parallel to the x-axis | (I) | \(\displaystyle\frac{x-1}{1}=\frac{y+1}{2}=\frac{z-2}{2}\) |
| (B) | passing through the origin | (II) | \(\displaystyle\frac{x-1}{1}=\frac{y+1}{0}=\frac{z-2}{0}\) |
| (C) | parallel to the line joining (3, 0, 1) and (4, 2, 3) | (III) | \(\displaystyle\frac{x-1}{2}=\frac{y+1}{1}=\frac{z-2}{-2}\) |
| (D) | parallel to the line \(\vec{r} = 3\hat{\imath}+\lambda(2\hat{\imath}+\hat{\jmath}-2\hat{k})\) | (IV) | \(\displaystyle\frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(II), (B)-(IV), (C)-(I), (D)-(III)
Every equation in List-II passes through P(1, −1, 2), so each item is decided by its direction ratios, the denominators.
(A) parallel to the x-axis
This matches (II). The x-axis has direction ratios 1, 0, 0.
(B) passing through the origin
This matches (IV). The direction from O(0, 0, 0) to P(1, −1, 2) is
\(1-0,\ -1-0,\ 2-0 = 1, -1, 2\)
(C) parallel to the line joining (3, 0, 1) and (4, 2, 3)
This matches (I):
\(4-3,\ 2-0,\ 3-1 = 1, 2, 2\)
(D) parallel to the line \(\vec{r} = 3\hat{\imath}+\lambda(2\hat{\imath}+\hat{\jmath}-2\hat{k})\)
This matches (III). The direction vector \(2\hat{\imath}+\hat{\jmath}-2\hat{k}\) gives the direction ratios 2, 1, −2.