The line \(\displaystyle\frac{x-2}{1}=\frac{y-k}{2}=\frac{z-1}{-1}\) meets the z-axis. The value of k is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 4
Let each fraction be \(\lambda\). The general point of the line is
\((2+\lambda,\ k+2\lambda,\ 1-\lambda)\)
A point of the z-axis has x = 0 and y = 0:
\(2+\lambda = 0\), so \(\lambda = -2\)
\(k+2(-2) = 0\), so \(k = 4\)
The line then meets the z-axis at \((0, 0, 1+2) = (0, 0, 3)\).
Why the other options are incorrect
Option (1) −4 — Incorrect. With \(k = -4\), the point with x = 0 has \(y = -4-4 = -8 \ne 0\).
Option (2) 2 — Incorrect. With \(k = 2\), the point with x = 0 has \(y = 2-4 = -2 \ne 0\).
Option (3) 3 — Incorrect. 3 is the z-coordinate of the meeting point, not k; with \(k = 3\), \(y = 3-4 = -1 \ne 0\).