Let a and b be positive real numbers. Suppose P Q → = a i ^ + b j ^ and P S → = a i ^ - b j ^ are adjacent sides of a parallelogram P Q R S . Let u → and v → be the projection vectors of w → = i ^ + j ^ along P Q → and P S → , respectively. If | u → | + v → = w → and if the area of the parallelogram P Q R S is 8 , then which of the following statements is/are TRUE?
C. The length of the diagonal P R of the parallelogram P Q R S is 4
Projection of a → a l o n g b → = a → . b → b → · b ^ u → = i ^ + j ^ · a i ^ + b j ^ a 2 + b 2 a i ^ + b j ^ a 2 + b 2 = a + b a 2 + b 2 · a i ^ + b j ^ a 2 + b 2 v → = i ^ + j ^ · a i ^ - b j ^ a 2 + b 2 a i ^ - b j ^ a 2 + b 2 = a - b a 2 + b 2 · a i ^ - b j ^ a 2 + b 2 given i j k a b 0 a - b 0 = 8 k ^ - a b - a b = 8 a b = 4 Given u → + | v → | = w → a + b a 2 + b 2 a - b a 2 + b 2 = 2 For a > b , 2 a = 2 a 2 + b 2 2 a 2 = a 2 + b 2 a 2 = b 2 ⇒ a = b = 2 Similarly, for b > a , a = b = 2 a + b = 4 a - b = 0 Length of diagonal P R → = PS → + PQ → = 2 a i ^ = 2 a = 4 .
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