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

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

Vector Algebra

Solution:

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} $