Question

The distance of point $A(-2,3,1)$ from the line $PQ$ through $P(-3,5,2)$, which make equal angles with the axis is

• A. $2/\sqrt{3}$
• B. $\sqrt{14}/3$
• C. $16/\sqrt{3}$
• D. $5\sqrt{3}$

Answer 1

The correct option is A $2/\sqrt{3}$

Let the angles which PQ makes with the axis each be $\alpha$, then

$\begin{array}{c}{\cos }^2\alpha +{\cos }^2\alpha +{\cos }^2\alpha =1\\ \Rightarrow \cos \alpha =\frac{1}{\sqrt{3}}\\ \therefore Direction\cos inesofPQare\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)\\ \therefore ThelengthofprojectionofAPonPR\\ =\left(-2+3\right)\frac{1}{\sqrt{3}}+\left(3-5\right)\frac{1}{\sqrt{3}}+\left(1-2\right)\frac{1}{\sqrt{3}}\\ =\frac{-2}{\sqrt{3}}\\ \therefore A{R}^2=A{P}^2-P{R}^2\\ ={\left(1\right)}^2+{\left(-2\right)}^2+{\left(-1\right)}^2-{\left(\frac{-2}{\sqrt{3}}\right)}^2\\ =6-\frac{4}{3}=\frac{14}{3}\\ \therefore AR=\sqrt{\frac{14}{3}}\end{array}$