Let \( a \) and \( b \) be positive real numbers. Suppose \( \overrightarrow{P Q}=a \hat{i}+b \h...

Let \( a \) and \( b \) be positive real numbers. Suppose \( \overrightarrow{P Q}=a \hat{i}+b \hat{j} \) and \( \overrightarrow{P S}=a \hat{i}-b \hat{j} \) are adjacent sides of a parallelogram \( P Q R S \). Let \( \vec{u} \) and \( \vec{v} \) be the projection vectors of along \( \vec{w}=\hat{i}+\hat{j} \) along \( \overrightarrow{P Q} \) and \( \overrightarrow{P S} \), respectively. If \( |\vec{u}|+|\vec{v}|=|\vec{w}| \) and if the area of the parallelogram PQRS is 8 , then which of the following statements is/are TRUE?
(a) \( a+b=4 \)
(b) \( a-b=2 \)
(c) The length of the diagonal \( P R \) of the parallelogram \( P Q R S \) is 4
(d) \( \vec{w} \) is an angle bisector of the vectors \( \overrightarrow{P Q} \) and \( \overrightarrow{P S} \).