Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Line | Direction ratios | ||
| (A) | \(\displaystyle\frac{1-x}{2}=\frac{y+3}{1}=\frac{z}{3}\) | (I) | 1, −3, 0 |
| (B) | \(\displaystyle\frac{2x+1}{4}=\frac{3-y}{1}=\frac{z-2}{-2}\) | (II) | 6, 3, 2 |
| (C) | \(x = 2y-1 = 3z\) | (III) | −2, 1, 3 |
| (D) | \(\vec{r} = (2+\lambda)\hat{\imath}+(1-3\lambda)\hat{\jmath}+4\hat{k}\) | (IV) | 2, −1, −2 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
Direction ratios are read from the standard form \(\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\), in which x, y and z each have coefficient +1.
(A) \(\frac{1-x}{2}=\frac{y+3}{1}=\frac{z}{3}\)
This matches (III) −2, 1, 3:
\( \displaystyle \frac{1-x}{2} = \frac{x-1}{-2} \)
(B) \(\frac{2x+1}{4}=\frac{3-y}{1}=\frac{z-2}{-2}\)
This matches (IV) 2, −1, −2:
\( \displaystyle \frac{2x+1}{4} = \frac{x+\frac{1}{2}}{2} \quad\text{and}\quad \frac{3-y}{1} = \frac{y-3}{-1} \)
(C) \(x = 2y-1 = 3z\)
This matches (II) 6, 3, 2. Writing each part as a fraction with coefficient 1:
\( \displaystyle \frac{x}{1} = \frac{y-\frac{1}{2}}{\frac{1}{2}} = \frac{z}{\frac{1}{3}} \)
\( \displaystyle 1, \frac{1}{2}, \frac{1}{3} \ \text{times } 6 = 6, 3, 2 \)
(D) \(\vec{r} = (2+\lambda)\hat{\imath}+(1-3\lambda)\hat{\jmath}+4\hat{k}\)
This matches (I) 1, −3, 0. Separating the terms with \(\lambda\):
\(\vec{r} = (2\hat{\imath}+\hat{\jmath}+4\hat{k})+\lambda(\hat{\imath}-3\hat{\jmath}+0\hat{k})\)