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  4. Let the lines L 1: r =λ( hat i +2 hat j +3 hat k ), λ ∈ R L2: vecr=( hati+3 hatj+ hatk)+μ( hati+ hatj+5 hatk) ; μ ∈ R intersect at the point S. If a plane a x+b y-z +d=0 passes through S and is parallel to both the lines L1 and L2, then the value of a+b+ d is equal to

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Q. Let the lines
$L _{1}: \overrightarrow{ r }=\lambda(\hat{ i }+2 \hat{ j }+3 \hat{ k }), \lambda \in R$
$L_{2}: \vec{r}=(\hat{i}+3 \hat{j}+\hat{k})+\mu(\hat{i}+\hat{j}+5 \hat{k}) ; \mu \in R$
intersect at the point $S$. If a plane $a x+b y-z$ $+d=0$ passes through $S$ and is parallel to both the lines $L_{1}$ and $L_{2}$, then the value of $a+b+$ $d$ is equal to _________

JEE MainJEE Main 2022Three Dimensional Geometry

Solution:

Both the lines lie in the same plane
image
$\therefore$ equation of the plane
$\begin{vmatrix}x & y & z \\1 & 2 & 3 \\1 & 1 & 5\end{vmatrix}=0 $
$\Rightarrow 7 x-2 y-z=0 $
$\therefore a+b+d=5$