Let a and b be positive real numbers. Suppose PQ→=ai^+bj^ and PS→=ai^−bj^ are adjacent sides of a  parallelogram PQRS . Let u→ and v→ be the projection vectors of w→=i^+j^ along PQ→ and PS→ ,  respectively. If |u→|+|v→|=|w→| and if the area of the parallelogram PQRS is 8, then which of the  following statements is/are TRUE?

similarly ,u→=projection  vector of i^+j^on PQ, hence its length= |u→|=(i+j)⋅(ai+bj)a2+b2=a+ba2+b2 |v→|=(i+j)⋅(ai−bj)a2+b2=a−ba2+b2u→+|v→|=|w→||(a+b)|+|(a−b)|a2+b2=2 For a≥b2a=2⋅a2+b24a2=2a2+2b2a2=b2∴a=b …....(1)(a>0,b>0) similarly for b≥a  we will get a=b Now area of parallelogram =|(ai+bj)×(ai−bj)|=2ab∴2ab=8ab=4.......(2)from (1) and (2)a=2,b=2∴a+b=4  option (A)length of diagonal is |2ai|=|4i^|=4so option (C)