Consider the line \(\displaystyle\frac{4-x}{2}=\frac{3y+3}{6}=\frac{z-1}{-1}\).
(A) Its direction ratios are 2, 6, −1
(B) It passes through the point (4, −1, 1)
(C) \(\displaystyle -\frac{2}{3}, \frac{2}{3}, -\frac{1}{3}\) are direction cosines of the line
(D) It is parallel to the line \(\vec{r} = \hat{\imath}+\lambda(4\hat{\imath}+4\hat{\jmath}-2\hat{k})\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (C) only
The point and the direction ratios can be read only from the standard form \(\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\), where x, y and z each have coefficient 1.
\( \displaystyle \frac{4-x}{2} = \frac{x-4}{-2} \)
\( \displaystyle \frac{3y+3}{6} = \frac{3(y+1)}{6} = \frac{y+1}{2} \)
So the line is
\( \displaystyle \frac{x-4}{-2}=\frac{y+1}{2}=\frac{z-1}{-1} \)
\(\sqrt{4+4+1} = 3\)
So the line passes through (4, −1, 1), its direction ratios are −2, 2, −1, and its direction cosines are \(\pm\left(-\frac{2}{3}, \frac{2}{3}, -\frac{1}{3}\right)\).
Checking the statements
(A) Its direction ratios are 2, 6, −1 — Incorrect. The direction ratios are −2, 2, −1, and 2, 6, −1 is not a multiple of them.
(B) It passes through (4, −1, 1) — Correct. All three fractions are 0 at this point.
(C) \(-\frac{2}{3}, \frac{2}{3}, -\frac{1}{3}\) are direction cosines of the line — Correct. They are −2, 2, −1 divided by 3.
(D) It is parallel to \(\vec{r} = \hat{\imath}+\lambda(4\hat{\imath}+4\hat{\jmath}-2\hat{k})\) — Incorrect. 4, 4, −2 is not proportional to −2, 2, −1 (ratios −2, 2, 2).