The point on the line \(\displaystyle\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-4}{-1}\) whose z-coordinate is 6 is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (−4, −5, 6)
Let each ratio equal t. Then the general point of the line is
\((3t+2,\ 2t-1,\ -t+4)\)
The z-coordinate is 6:
\(-t+4 = 6\)
\(t = -2\)
\(x = 3(-2)+2 = -4\)
\(y = 2(-2)-1 = -5\)
So the point is (−4, −5, 6).
Why the other options are incorrect
Option (1) (8, 3, 6) — Incorrect. This uses t = 2, but t = 2 gives z = 2, not 6; the sign of the z-ratio −1 was missed.
Option (3) (−1, −3, 6) — Incorrect. These x and y belong to t = −1, which gives z = 5.
Option (4) (5, 1, 6) — Incorrect. These x and y belong to t = 1, which gives z = 3.