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 $P Q$ are
$\lambda-1,2 \lambda+1-6,3 \lambda+2-3 $
$\Rightarrow \lambda-1,2 \lambda-5,3 \lambda-1 $
$\because P Q$ 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}} $
$=\sqrt{9+4}=\sqrt{13} .$