CBSE Class 12 Mathematics 2026: question 35

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Step-by-step solution

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1. The direction ratios of the first line are $⟨2,3,4⟩$. Writing the second line as $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z-0}{1}$, its direction ratios are $⟨5,2,1⟩$.

   1 mark

2. Since $\frac{2}{5}\ne \frac{3}{2}\ne \frac{4}{1}$, the direction ratios are not proportional. Hence the lines are not parallel.

   1 mark

3. A general point on line 1 is $(2\lambda +1,3\lambda +2,4\lambda +3)$.
   A general point on line 2 is $(5\mu +4,2\mu +1,\mu )$.

   1 mark

4. For the lines to intersect, these points must coincide for some $\lambda ,\mu$:
   $2\lambda +1=5\mu +4$, $3\lambda +2=2\mu +1$, $4\lambda +3=\mu$.

   The first two give $2\lambda -5\mu =3$ and $3\lambda -2\mu =-1$, so $\lambda =-1$, $\mu =-1$.

   Check in the third equation: $4(-1)+3=-1=\mu$. It is satisfied, so the lines intersect.

   1 mark

5. Put $\lambda =-1$ in the point on line 1: $(2(-1)+1,3(-1)+2,4(-1)+3)$.
   The point of intersection is $(-1,-1,-1)$.

   1 mark

Answer: The lines are not parallel; they intersect at $(-1,-1,-1)$.

Revise the chapter

• Chapter: Three Dimensional Geometry (notes, formulas, where marks are lost and every past question)
• Topic: Shortest Distance Between Two Lines