Question

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

• A. $\frac{\sqrt{5}}{3}$
• B. $\frac{\sqrt{7}}{3}$
• C. $\frac{2}{3}$
• D. $3$

Answer 1

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

Since the line is perpendicular to the plane $x+2y+2z=0$, so directon of line will be $(1,2,2)$ and it passes through $(0,1,0)$,
Therefore equation of line is $\frac{x}{1}=\frac{y-1}{2}=\frac{z}{2}$
So general point on the line is $(r,2r+1,2r)$
Now direction ratio of line passing through this point and origin which perpendicular to given line is $(r,2r+1,2r)$
Since they are perpendicular, so dot product will be $0$ .
$r+2\times (2r+1)+2\times 2r=0$
$r+4r+2+4r=0$
$\therefore r=$$\frac{-2}{9}$
So points on line which perpendicular from origin is $(-2/9,5/9,-4/9)$
Now applying distance formula,
$=$ $\sqrt{{(-2/9)}^2+{\left(5/9\right)}^2+{(-4/9)}^2}$
$=$ $\sqrt{4/81+25/81+16/81}$
$=$ $\frac{\sqrt{5}}{3}$