Question

Let $L_1:\frac{x-1}{1}=\frac{y-1}{2}=\frac{z-1}{k}$ and $L_2:\frac{x-3}{1}=\frac{y-2}{-1}=\frac{z-3}{-1}$ be two intersecting lines and $L_3$ be a straight line which is perpendicular to $L_1$ and $L_2$ and passing through their point of intersection. If length of perpendicular from $(3,1,2)$ to the line $L_3$ is $\lambda$ then $[\lambda ]$ equals
[Note: [y] denotes the greatest integer function less than or equal to $y$.]

Answer 1

Answer: 2

$L_1$ and $L_2$ are intersecting $\Rightarrow k=3$ and their point of intersection is $(2,3,4)$.
$L_3:\frac{x-2}{1}=\frac{y-3}{4}=\frac{z-4}{-3}$
$P=\frac{1(\hat{i}-2\hat{j}-2\hat{k})\times (\hat{i}+4\hat{j}-3\hat{k})}{|\hat{i}+4\hat{j}-3\hat{k}|}=\frac{\sqrt{233}}{\sqrt{26}}$

$[P]=[\sqrt{8\cdot 9}=2.$