The lines \(\displaystyle\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{2}\) and \(\displaystyle\frac{x-5}{-4}=\frac{y+3}{2}=\frac{z-7}{-4}\) are
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → coincident
The direction ratios are 2, −1, 2 and −4, 2, −4. Since
\(-4, 2, -4 = -2(2, -1, 2)\)
the lines are parallel. So they are either distinct or the same line; this depends on whether a point of one lies on the other.
The second line passes through (5, −3, 7). Putting it in the first line:
\( \displaystyle \frac{5-1}{2} = 2,\quad \frac{-3+1}{-1} = 2,\quad \frac{7-3}{2} = 2 \)
All three are equal, so (5, −3, 7) lies on the first line too. Two parallel lines with a common point are the same line.