The equation of the line passing through the point (2, 0, −1) and parallel to the line \(\displaystyle\frac{4-x}{2}=\frac{y+1}{3}=\frac{2z-1}{4}\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{x-2}{-2}=\frac{y}{3}=\frac{z+1}{2}\)
The direction ratios of the given line are read from its standard form:
\( \displaystyle \frac{4-x}{2} = \frac{x-4}{-2} \)
\( \displaystyle \frac{2z-1}{4} = \frac{z-\frac{1}{2}}{2} \)
So the given line has direction ratios −2, 3, 2. A parallel line has the same direction ratios. Through (2, 0, −1):
\( \displaystyle \frac{x-2}{-2}=\frac{y-0}{3}=\frac{z+1}{2} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{x-2}{2}=\frac{y}{3}=\frac{z+1}{4}\) — Incorrect. This copies the printed denominators 2, 3, 4 without making the coefficients of x and z equal to 1.
Option (2) \(\displaystyle\frac{x+2}{-2}=\frac{y}{3}=\frac{z-1}{2}\) — Incorrect. The direction is right, but this line passes through (−2, 0, 1), not (2, 0, −1).
Option (3) \(\displaystyle\frac{x-2}{-2}=\frac{y}{3}=\frac{z+1}{4}\) — Incorrect. The coefficient 2 of z was not divided out; −2, 3, 4 is not proportional to −2, 3, 2.