The image of the point P(3, 2, 1) in the line \(\displaystyle\frac{x}{1}=\frac{y-1}{2}=\frac{z+1}{2}\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (−1, 4, 1)
The image Q is placed so that the foot of the perpendicular M from P to the line is the midpoint of PQ.
The general point of the line is
\(M = (t,\ 2t+1,\ 2t-1)\)
\(\overrightarrow{MP} = (3-t,\ 1-2t,\ 2-2t)\)
\(\overrightarrow{MP}\) is perpendicular to the direction 1, 2, 2:
\((3-t)+2(1-2t)+2(2-2t) = 0\)
\(9-9t = 0\)
\(t = 1\)
So M = (1, 3, 1). Since M is the midpoint of PQ, Q = 2M − P:
\(Q = (2-3,\ 6-2,\ 2-1) = (-1, 4, 1)\)
Why the other options are incorrect
Option (1) (1, 3, 1) — Incorrect. This is the foot M, the midpoint of P and its image, not the image itself.
Option (3) (5, 1, 1) — Incorrect. This is P + (P − M), which moves away from the line instead of across it.
Option (4) (−3, −2, −1) — Incorrect. This is the image of P in the origin, not in the line.