Question

If $(2,-1,3)$ is the foot of the perpendicular drawn from the origin to the plane, then the equation of the plane is

• A. $2x+y-3z+6=0$
• B. $2x-y+3z-14=0$
• C. $2x-y+3z-13=0$
• D. None of these

Answer 1

The correct option is B $2x-y+3z-14=0$

Let $P(2,-1,3)$ be the foot of the perpendicular from the origin to the plane.

$\therefore \overrightarrow{OP}=\overrightarrow{n}=2\overrightarrow{i}-\overrightarrow{j}+3\overrightarrow{k}$ and $p=|2\overrightarrow{i}-\overrightarrow{j}+3\overrightarrow{k}|=\sqrt{2^2+1^2+3^2}=\sqrt{4+1+9}=\sqrt{14}$

$\therefore \hat{n}=\frac{2\overrightarrow{i}-\overrightarrow{j}+3\overrightarrow{k}}{\sqrt{14}}$

The vector equation of the plane is $\overrightarrow{r}\cdot \hat{n}=p$

$\Rightarrow (x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k})\cdot \frac{2\overrightarrow{i}-\overrightarrow{j}+3\overrightarrow{k}}{\sqrt{14}}=\sqrt{14}$

$\Rightarrow (x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k})\cdot (2\overrightarrow{i}-\overrightarrow{j}+3\overrightarrow{k})=14$

$\Rightarrow 2x-y+3z=14$

$\Rightarrow 2x-y+3z-14=0$