1. Tardigrade
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  4. Let a plane P contain two lines vec r = hat i +λ( hat i + hat j ), λ ∈ R and vec r =- hat j +μ( hat j - hat k ), μ ∈ R If Q (α, β, γ) is the foot of the perpendicular drawn from the point M (1,0,1) to P , then 3(α+β+γ) equals.

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Q. Let a plane $P$ contain two lines $\vec{ r }=\hat{ i }+\lambda(\hat{ i }+\hat{ j }), \lambda \in R$ and $\vec{ r }=-\hat{ j }+\mu(\hat{ j }-\hat{ k }), \mu \in R$
If $Q (\alpha, \beta, \gamma)$ is the foot of the perpendicular drawn from the point $M (1,0,1)$ to $P $, then $3(\alpha+\beta+\gamma)$ equals____.

JEE MainJEE Main 2020Three Dimensional Geometry

Solution:

Dr's normal to plane
$=\begin{vmatrix}i&j&k\\ 1&1&0\\ 0&1&-1\end{vmatrix}=\hat{i}+\hat{j}+\hat{k}$
Equation of plane
$-1(x-1)+1(y-0)+1(z-0)=0$
$x-y-z-1=0\,\,$.....(1)
Now $\frac{\alpha-1}{1}=\frac{\beta-0}{-1}=\frac{\gamma-1}{-1}=-\frac{(1-0-1-1)}{3}$
$\frac{\alpha-1}{1}=\frac{\beta}{-1}=\frac{\gamma-1}{-1}=\frac{1}{3}$
$\alpha=\frac{4}{3}, \beta=-\frac{1}{3}, \gamma=\frac{2}{3}$
$3(\alpha+\beta+\gamma)=3\left(\frac{4}{3}-\frac{1}{3}+\frac{2}{3}\right)=5$