The points on the line \(\displaystyle\frac{x-1}{2}=\frac{y+2}{-2}=\frac{z-3}{1}\) which are at a distance of 6 units from the point A(1, −2, 3) are
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (5, −6, 5) and (−3, 2, 1)
A is on the line (all three fractions are 0 there). The direction ratios 2, −2, 1 have length
\(\sqrt{4+4+1} = 3\)
so the direction cosines are \(\frac{2}{3}, -\frac{2}{3}, \frac{1}{3}\). Moving a distance 6 along the line in either direction from A:
\( \displaystyle 6 \times \left(\frac{2}{3}, -\frac{2}{3}, \frac{1}{3}\right) = (4, -4, 2) \)
\((1+4, -2-4, 3+2) = (5, -6, 5)\)
\((1-4, -2+4, 3-2) = (-3, 2, 1)\)
Why the other options are incorrect
Option (2) (3, −4, 4) and (−1, 0, 2) — Incorrect. These points are on the line but only 3 units from A (they come from adding the direction ratios once).
Option (3) (13, −14, 9) and (−11, 10, −3) — Incorrect. These come from 6 times the direction ratios instead of the direction cosines; they are 18 units from A.
Option (4) (7, 4, 9) and (−5, −8, −3) — Incorrect. For (7, 4, 9), \(\frac{7-1}{2} = 3\) but \(\frac{4+2}{-2} = -3\), so it is not on the line.