A line passes through (2, 1, 3) and (1, 2, -1), then
(A) Equation of the line is \( \displaystyle\frac{x-2}{-1}=\displaystyle\frac{y-1}{1}=\displaystyle\frac{z-3}{-4} \).
(B) Equation of the line is \( \displaystyle\frac{x+2}{-1}=\displaystyle\frac{y+1}{1}=\displaystyle\frac{z+3}{-4} \).
(C) Equation of the line is \( \vec{r}=2\hat{\imath}+\hat{\jmath}+3\hat{k}+\lambda(\hat{\imath}-\hat{\jmath}+4\hat{k}) \).
(D) Equation of the line is \( \displaystyle\frac{x-1}{1}=\displaystyle\frac{y-2}{-1}=\displaystyle\frac{z+1}{4} \).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
Direction ratios of the line through (2, 1, 3) and (1, 2, −1):
\( (1-2,\ 2-1,\ -1-3)=(-1,\ 1,\ -4) \)
Any multiple, such as (1, −1, 4), gives the same line, and any point of the line may be used.
So a correct equation must pass through a point of the line, such as (2, 1, 3) or (1, 2, −1), with direction (−1, 1, −4) or a multiple of it.
Checking the statements
(A) Equation of the line is \( \displaystyle\frac{x-2}{-1}=\displaystyle\frac{y-1}{1}=\displaystyle\frac{z-3}{-4} \). — Correct. It passes through (2, 1, 3) with direction (−1, 1, −4).
(B) Equation of the line is \( \displaystyle\frac{x+2}{-1}=\displaystyle\frac{y+1}{1}=\displaystyle\frac{z+3}{-4} \). — Incorrect. It passes through (−2, −1, −3), which is not on the given line.
(C) Equation of the line is \( \vec{r}=2\hat{\imath}+\hat{\jmath}+3\hat{k}+\lambda(\hat{\imath}-\hat{\jmath}+4\hat{k}) \). — Correct. It passes through (2, 1, 3), and (1, −1, 4) is a multiple of (−1, 1, −4).
(D) Equation of the line is \( \displaystyle\frac{x-1}{1}=\displaystyle\frac{y-2}{-1}=\displaystyle\frac{z+1}{4} \). — Correct. It passes through (1, 2, −1) with direction (1, −1, 4).