CBSE 2026 · Region 1 · Set 1 · Q35 · 5 marks
Check whether the lines given by $\displaystyle \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\displaystyle \frac{x-4}{5}=\frac{y-1}{2}=z$ are parallel or not. If parallel, find the distance between them, otherwise find their point of intersection, if the lines are intersecting.
Official answer
From CBSE’s own marking scheme for this paper.
The lines are not parallel. They intersect at the point ($\displaystyle 3$, $\displaystyle 5$, $\displaystyle 7$).
Marking-scheme solution
DRs of first line: $\displaystyle \langle2,3,4\rangle$
DRs of second line: $\displaystyle \langle5,2,1\rangle$
DRs of the two lines are not proportional. Hence, the lines are not parallel.
Any Point on first line: $\displaystyle (2\lambda+1, 3\lambda+2, 4\lambda+3)$;
Any point on second line: $\displaystyle (5\mu+4, 2\mu+1, \mu)$
For the lines to intersect, we must have some $\displaystyle \lambda$ and $\displaystyle \mu$ for which these coordinates coincide, i.e. $\displaystyle 2\lambda+1=5\mu+4$, $\displaystyle 3\lambda+2=2\mu+1$, $\displaystyle 4\lambda+3=\mu$, for some $\displaystyle \lambda$ and $\displaystyle \mu$
Solving the first two coordinate equations: $\displaystyle \lambda=-1$, $\displaystyle \mu=-1$; this satisfies the third equation too.
The lines intersect at $\displaystyle (-1,-1,-1)$
Three Dimensional GeometryShortest Distance between Two LinesApplylong_answermedium
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CBSE Class 12 Mathematics past-paper question from the 2026board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.