Question

Let $P$ be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes $\overrightarrow{r}.(\hat{i}+\hat{j}+4\hat{k})$ and $\overrightarrow{r}.(-\hat{i}+\hat{j}+\hat{k})=6.$
Then which of the following points does NOT lie on $P?$

• A. $(-8,8,6)$
• B. $(4,2,2)$
• C. $(3,3,2)$
• D. $(6,-6,2)$

Answer 1

The correct option is B $(4,2,2)$

Equation of plane through point of intersection of planes
$\overrightarrow{r}.(\hat{i}+\hat{j}+4\hat{k})=16$ and $\overrightarrow{r}.(-\hat{i}+\hat{j}+\hat{k})=6$ is
$(x+y+4z-16)+\lambda (-x+y+z-6)=0\cdots \left(i\right)$
This plane passes through point $(1,2,3),$ then
$-1-2\lambda =0$
$\therefore \lambda =-\frac{1}{2}$
$\therefore$ Equation of plane is:
$2x+2y+8z-32+x-y-z+6=0$
$\therefore 3x+y+7z-26=0$

Clearly, $(4,2,2)$ does not lies on the plane.