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  4. If the foot of the perpendicular drawn from (1,9,7) to the line passing through the point (3,2,1) and parallel to the planes x+2 y+z=0 and 3 y-z=3 is (α, β, γ), then α+β+γ is equal to

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Q. If the foot of the perpendicular drawn from $(1,9,7)$ to the line passing through the point $(3,2,1)$ and parallel to the planes $x+2 y+z=0$ and $3 y-z=3$ is $(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma$ is equal to

JEE MainJEE Main 2023Three Dimensional Geometry

Solution:

Direction ratio of line $= \begin{vmatrix} \hat{ i } & \hat{ j } & \hat{ k } \\ 1 & 2 & 1 \\ 0 & 3 & -1\end{vmatrix} $
$ =\hat{ i }(-5)-\hat{ j }(-1)+\hat{ k }(3) $
$ =-5 \hat{ i }+\hat{ j }+3 \hat{ k }$
image
$ M (-5 \lambda+3, \lambda+2,3 \lambda+1) $
$ \overrightarrow{\operatorname{PM}} \perp(-5 \hat{ i }+\hat{ j }+3 \hat{ k }) $
$ -5(-5 \lambda+2)+(\lambda-7)+3(3 \lambda-6)=0 $
$ \Rightarrow 25 \lambda+\lambda+9 \lambda-10-7-18=0 $
$ \Rightarrow \lambda=1 $
Point $ M =(-2,3,4)=(\alpha, \beta, \gamma)$
$\alpha+\beta+\gamma=5$