Question

$OM$ is the perpendicular from $O(0,0,0)$ to a plane, if $|\overrightarrow{OM}|$ $=$ $3$ and $2,2,1$ are the D.R.s of $OM$, then the $z-$ intercept of the plane is :

• A. $9$
• B. $-9$
• C. $3$
• D. $-3$

Answer 1

The correct option is A $9$

General equation of plane at a distance $OM$ from the origin is given by,

$\overrightarrow{r}\cdot \hat{n}=OM$, where $\hat{n}$ is unit vector along $OM$

Given direction ratios of $OM$ are $2,2,1$, so

$\hat{n}=\frac{2}{3}\hat{i}+\frac{2}{3}\hat{j}+\frac{1}{3}\hat{k}$

Putting $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$ and $|OM|=3$, equation of plane becomes,

$2x+2y+z=9$

Hence, $z$-intercept of this plane is $9$