Which of the following points lies on the line \(\displaystyle\frac{x+1}{2}=\frac{y-3}{-1}=\frac{z-2}{4}\)?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (1, 2, 6)
Putting each fraction equal to \(\lambda\) gives the general point of the line:
\(x = -1+2\lambda,\quad y = 3-\lambda,\quad z = 2+4\lambda\)
For (1, 2, 6), the x-coordinate gives \(\lambda = 1\). Then
\(y = 3-1 = 2,\quad z = 2+4 = 6\)
Both agree, so (1, 2, 6) lies on the line.
Why the other options are incorrect
Option (1) (1, 4, 6) — Incorrect. x = 1 needs \(\lambda = 1\), which gives y = 2, not 4.
Option (3) (−3, 2, −2) — Incorrect. x = −3 needs \(\lambda = -1\), which gives y = 4, not 2.
Option (4) (3, 1, 6) — Incorrect. x = 3 needs \(\lambda = 2\), which gives z = 10, not 6.