Let a and b be positive real numbers. Suppose {PQ} = {a}{i} + {b}{j} and {PS} = {a}{i} - {b}{j} are adjacent sides of aparallelogram 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? [JEE Adv 2020 P2]

a + b = 4
a – b = 2
The length of the diagonal PR of the parallelogram PQRS is 4
$\vec{w}$ is an angle bisector of the vectors $\overrightarrow{PQ}$ and $\vec{PS}$

Solution:

|\vec{v}| = \frac{|a+b|}{\sqrt{a^2+b^2}}

and

|\vec{u}| = \frac{|a-b|}{\sqrt{a^2+b^2}}

|\vec{v}| + |\vec{u}| = |\vec{w}|

\Rightarrow \frac{|a+b|+|a-b|}{\sqrt{a^2+b^2}} = \sqrt{2}

\text{WLOG}, a \ge b

\sqrt{2} a = \sqrt{a^2+b^2}

\Rightarrow a = b

Also area of PQRS

= |\vec{PQ} \times \vec{PS}| = 2ab = 8

a = b = 2,

Also

\vec{PQ} \cdot \vec{PS} = a^2 - b^2 = 0

So PQRS is a square of side length

2 \sqrt{2}

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