Question

A line is perpendicular to plane $x+2y+2z=0$ and passes through $(0,1,0)$. The perpendicular distance of this line from origin is

• A. $\frac{\sqrt{5}}{3}$
• B. $\frac{3}{\sqrt{5}}$
• C. $\frac{\sqrt{3}}{5}$
• D. $\frac{5}{\sqrt{3}}$

Answer 1

The correct option is A $\frac{\sqrt{5}}{3}$

Equation of line is $\frac{x}{1}=\frac{y-1}{2}=\frac{z}{2}$
$\overrightarrow{r}=\hat{j}+\lambda (\hat{i}+2\hat{j}+2\hat{k})$
Hence, required distance is

$AP=|\overrightarrow{AB}\times \left(\frac{\hat{i}+2\hat{j}+2\hat{k}}{3}\right)|$

$=|\hat{j}\times \left(\frac{\hat{i}+2\hat{j}+2\hat{k}}{3}\right)|$

$=\left|\frac{-\hat{k}+2\hat{i}}{3}\right|=\frac{\sqrt{5}}{3}$

Hence, option A is correct.