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  4. The distance of the point P (4,6,-2) from the line passing through the point (-3,2,3) and parallel to a line with direction ratios 3,3,-1 is equal to :

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Q. The distance of the point $P (4,6,-2)$ from the line passing through the point $(-3,2,3)$ and parallel to a line with direction ratios $3,3,-1$ is equal to :

JEE MainJEE Main 2023Three Dimensional Geometry

Solution:

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Equation of line is $\frac{ x +3}{3}=\frac{ y -2}{3}=\frac{ z -3}{-1}=\lambda$
$M (3 \lambda-3,3 \lambda+2,3-\lambda)$
D.R of PM( $3 \lambda-7,3 \lambda-4,5-\lambda)$
Since PM is perpendicular to line
$ \Rightarrow 3(3 \lambda-7)+3(3 \lambda-4)-1(5-\lambda)=0 $
$ \Rightarrow \lambda=2$
$ \Rightarrow M (3,8,1) \Rightarrow PM =\sqrt{14}$