50. Given PQ=ai+bj and PS =ai-bj are adjacent sides of a parallelogram PQRS,so the area of parallelgram PQRS = $\left|PQ\times RS\right|=2ab=8\Rightarrow ab=4------\left(1\right)$  
According to question $\left|u\right|\left.=\right|$ projection vector of w =i+j along $PQ$

$=\left|\frac{\left(i+j\right).\left(ai+bj\right)}{\sqrt{a^2+b^2}}\right|=\left|\frac{a+b}{\sqrt{a^2+b^2}}\right|$    
similarly $\left|v\right|\left.=\right|$ projection of w =i+j along  $\left.ps\right|=\frac{a-b}{\sqrt{a^2+b^2}}$
$\begin{array}{l}\left|u\right|+\left|v\right|=\left|w\right|\Rightarrow \left|\frac{a+b}{\sqrt{a^2+b^2}}\right|=\sqrt{2}\\ ifa\ge 0b\ge 0then4a^2=2a^2+2b^2\Rightarrow a=b-----\left(2\right)\\ from\left(1\right)and\left(2\right)a=b=2\Rightarrow a+b=4\end{array}$
and the length of diagnol PR 
$\left|ai+bj+ai-bj\right|=\left|2ai\right|\Rightarrow 2a=4$
and the angular bisector of PQ and PS is along the vector  $\pm \frac{\lambda }{\sqrt{a^2+b^2}}\left(\left(ai+bj\right)\pm \left(ai-bj\right)\right)$