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  4. Let Q be the mirror image of the point P (1,2,1) with respect to the plane x +2 y +2 z =16. Let T be a plane passing through the point Q and contains the line vecr=- hatk+λ( hati+ hatj+2 hatk), λ ∈ R. Then, which of the following points lies on T?

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Q. Let $Q$ be the mirror image of the point $P (1,2,1)$ with respect to the plane $x +2 y +2 z =16$. Let $T$ be a plane passing through the point $Q$ and contains the line $\vec{r}=-\hat{k}+\lambda(\hat{i}+\hat{j}+2 \hat{k}), \lambda \in R$. Then, which of the following points lies on $T$?

JEE MainJEE Main 2022Three Dimensional Geometry

Solution:

Image of $P (1,2,1)$ in $x +2 y +2 z -16=0$
is given by $Q (4,8,7)$
Eq. of plane $T =\begin{vmatrix} x & y & z +1 \\ 4 & 8 & 6 \\ 1 & 1 & 2\end{vmatrix}=0$
$\Rightarrow 2 x - z =1$ so $B (1,2,1)$ lies on it.