Line and Plane — Maths STD 12 Science — Question

(c) : Given lines are
$\begin{aligned}
& \frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}=p \text { (say) } ...(i) \\
& \text { and } \frac{x-3}{1}=\frac{y-\lambda}{2}=\frac{z}{1}=\mu \text { (say) } ...(ii)
\end{aligned}$
$\therefore \quad$ The coordinates of a point on the line (i) are
$(2 p+1,3 p-1,4 p+1)$
The coordinates of a point on the line (ii) are
$(\mu+3,2 \mu+\lambda, \mu)$
Since two lines intersect each other.
$\begin{aligned}
\therefore \quad 2 p+1 & =\mu+3 ...(iii) \\
3 p-1 & =2 \mu+\lambda ...(iv) \\
4 p+1 & =\mu ...(v)
\end{aligned}$
Substituting value of $\mu$ from (v) in (iii) and (iv), we get
$\begin{aligned}
& 2 p+1=4 p+1+3 \Rightarrow 2 p=-3 \\
& \Rightarrow p=-3 / 2
\end{aligned}$
And $3 p-1=2(4 p+1)+\lambda$
$\begin{aligned}
& \Rightarrow 3\left(-\frac{3}{2}\right)-1=2\left(4 \times\left(\frac{-3}{2}\right)+1\right)+\lambda \\
& \Rightarrow \frac{-9}{2}-1=2(-6+1)+\lambda \Rightarrow \lambda=\frac{9}{2}
\end{aligned}$