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  4. The equation of the line through the point (0,1,2) and perpendicular to the line (x-1/2)=(y+1/3)=(z-1/-2) is

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Q. The equation of the line through the point (0,1,2) and perpendicular to the line
$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is

JEE MainJEE Main 2021Three Dimensional Geometry

Solution:

$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}=r$
$\Rightarrow P(x, y, z)=(2 r+1,3 r-1,-2 r+1)$
Since, $\overrightarrow{QP} \perp(2 \hat{i}+3 \hat{j}-2 \hat{k})$
$\Rightarrow 4 r+2+9 r-6+4 r+2=0$
$\Rightarrow r=\frac{2}{17}$
image
$\Rightarrow P\left(\frac{21}{17}, \frac{-11}{17}, \frac{13}{17}\right)$
$\Rightarrow \overrightarrow{PQ}=\frac{21 \hat{i}-28 \hat{j}-21 \hat{k}}{17}$
So, $\overrightarrow{QP}: \frac{x}{-3}=\frac{y-1}{4}=\frac{z-2}{3}$