Let P be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes
Question:

Let $P$ be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes

$\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}+4 \hat{\mathrm{k}})=16$ and $\overrightarrow{\mathrm{r}} \cdot(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})=6 .$ Then

which of the following points does NOT lie on P?

1. $(3,3,2)$

2. $(6,-6,2)$

3. $(4,2,2)$

4. $(-8,8,6)$

Correct Option: , 3

Solution:

$(x+y+4 z-16)+\lambda(-x+y+z-6)=0$

Passes through $(1,2,3)$

$-1+\lambda(-2) \Rightarrow \lambda=-\frac{1}{2}$

$2(x+y+4 z-16)-(-x+y+z-6)=0$

$3 x+y+7 z-26=0$