Question

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

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

Answer 1

Answer: (B)

The equation of line is $\frac{x-0}{1}=\frac{y-1}{2}=\frac{z-0}{2}=\lambda$ (say). $P(\lambda ,2\lambda +1,2\lambda )$
Let $D(\lambda )=\sqrt{{\lambda }^2+(2\lambda +1{)}^2+(2\lambda {)}^2};$ Put $D^{\prime }(\lambda )=0\Rightarrow \lambda =\frac{-2}{9}$
$\therefore D_{\min }\left(\lambda =\frac{-2}{9}\right)=\frac{\sqrt{45}}{9}=\frac{\sqrt{5}}{3}$