CBSE 2026 · Region 2 · Set 1 · Q35 · 5 marks
A line passing through the points A($\displaystyle 1$, $\displaystyle 2$, $\displaystyle 3$) and B($\displaystyle 5$, $\displaystyle 8$, $\displaystyle 11$) intersects the line $\displaystyle \vec{r}=4 \hat{i}+\hat{j}+\lambda(5 \hat{i}+2 \hat{j}+\hat{k})$. Find the co-ordinates of the point of intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines.
Marking-scheme solution
The line passing through the points $\displaystyle A(1,2,3)$ and $\displaystyle B(5,8,11)$ is given by
$\displaystyle l_{1}: \dfrac{x-1}{5-1}=\dfrac{y-2}{8-2}=\dfrac{z-3}{11-3}$ i.e. $\displaystyle \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}=\mu$
Also the given line in its cartesian form is, $\displaystyle l_{2}: \dfrac{x-4}{5}=\dfrac{y-1}{2}=\dfrac{z}{1}=\lambda$
Any point on $\displaystyle l_{1}$ is $\displaystyle P(2\mu+1, 3\mu+2, 4\mu+3)$
Any point on $\displaystyle l_{2}$ is $\displaystyle Q(5\lambda+4, 2\lambda+1, \lambda)$
When the lines intersect, the points $\displaystyle P$ and $\displaystyle Q$ must coincide.
$\displaystyle \therefore 2\mu+1=5\lambda+4, 3\mu+2=2\lambda+1$ and $\displaystyle 4\mu+3=\lambda$
$\displaystyle \Rightarrow 2\mu-5\lambda=3$ ..... (i) , $\displaystyle 3\mu-2\lambda=-1$ ...... (ii) and $\displaystyle 4\mu-\lambda=-3$ ...... (iii)
solving any two above equations, we get $\displaystyle \mu=\lambda=-1$
The point of intersection is $\displaystyle (-1,-1,-1)$.
Let the required line $\displaystyle (l)$ passing through $\displaystyle (-1,-1,-1)$ be $\displaystyle \dfrac{x+1}{a}=\dfrac{y+1}{b}=\dfrac{z+1}{c}$
Since $\displaystyle l \perp l_{1}$ and $\displaystyle l \perp l_{2}$,
$\displaystyle 2a+3b+4c=0, 5a+2b+c=0$
$\displaystyle \Rightarrow \dfrac{a}{3-8}=\dfrac{b}{20-2}=\dfrac{c}{4-15}$ i.e. $\displaystyle \dfrac{a}{-5}=\dfrac{b}{18}=\dfrac{c}{-11}$
Hence, the required line is $\displaystyle \dfrac{x+1}{-5}=\dfrac{y+1}{18}=\dfrac{z+1}{-11}$ or $\displaystyle \dfrac{x+1}{5}=\dfrac{y+1}{-18}=\dfrac{z+1}{11}$
Three Dimensional GeometryEquation of a Line in SpaceApplylong_answerhard
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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.