If the length of the projection of the line segment with points (1, 0, - 1) and (-1, 2,2) to the plane x + 3y - 5z=6 is4 then find the value of  $\left[\frac{d}{2}\right]$, where [ . ] represent greatest integer function.

Let A(1, 0, -1), B(-1 ,2,2) Direction ratios of AB are (2, - 2, - 3) 

Let $\theta$ be the angle between the line and normal to plane; then 

$\begin{array}{r}\cos \mathrm{\theta }=\frac{|2.1+3(-2)-5(-3\left)\right|}{\sqrt{1+9+25}\sqrt{4+4+9}}\\ =\frac{11}{\sqrt{17}\sqrt{35}}=\frac{11}{\sqrt{595}}\end{array}$

Length of projection 

$\begin{array}{l}=\left(\mathrm{AB}\right)\sin \mathrm{\theta }\\ =\sqrt{(2{)}^2+(2{)}^2+(3{)}^2}\times \sqrt{1-\frac{121}{595}}\\ =\sqrt{\frac{474}{35}}\text{ units }\end{array}$