Question

The length of perpendicular from $(1,6,3)$ to the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ is:

• A. $3$
• B. $\sqrt{11}$
• C. $\sqrt{13}$
• D. $5$

Answer 1

Answer: (C)

Given, line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$
$\Rightarrow \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}=\lambda$ (say)

$\Rightarrow x=\lambda ,y=2\lambda +1,z=3\lambda +2$
Therefore, direction ratios of $PQ$ are
$\lambda -1,2\lambda +1-6,3\lambda +2-3$
$\Rightarrow \lambda -1,2\lambda -5,3\lambda -1$
$\because PQ$ is perpendicular to the given line. Therefore,
$1(\lambda -1)+2(2\lambda -5)+3(3\lambda -1)=0$
$\Rightarrow \lambda =1$
$\therefore$ The coordinate of $Q(1,3,5)$
$\therefore$ Length of perpendicular
$=\sqrt{(1-1{)}^2+(3-6{)}^2+(5-3{)}^2}$
$=\sqrt{9+4}=\sqrt{13}.$