Question

The length of the perpendicular from $\left(1,0 , 2\right)$ on the line $\frac{x + 1}{3}=\frac{y - 2}{- 2}=\frac{z + 1}{- 1}$ is

• A. $\frac{3 \sqrt{6}}{2}$ units
• B. $\frac{6 \sqrt{3}}{5}$ units
• C. $3\sqrt{2}$ units
• D. $2\sqrt{3}$ units

Answer 1

Answer: (A)

Direction ratios of $PR \rightarrow 3\lambda -2,-2\lambda +2,-\lambda -3$
Direction Ratios of the line $QR \rightarrow 3,-2,-1$
Since, they are perpendicular
$3\left(3 \lambda - 2\right)-2\left(- 2 \lambda + 2\right)-1\left(- \lambda - 3\right)=0$
$9\lambda -6+4\lambda -4+\lambda +3=0$
$14\lambda =7$
$\lambda =\frac{1}{2}$
$R \rightarrow \left(\frac{1}{2} , 1 , - \frac{3}{2}\right)$
$PR=\sqrt{\left(1 - \frac{1}{2}\right)^{2} + \left(0 - 1\right)^{2} + \left(2 + \frac{3}{2}\right)^{2}}$
$=\sqrt{\frac{1}{4} + \frac{4}{4} + \frac{49}{4}}$
$=\sqrt{\frac{54}{4}}$
$=\frac{3 \sqrt{6}}{2}$ units