KVPY SA Exam 2018 Question Paper

Let A (0,0), B(1,0), C(1,1) & D(0, 1) ⇒ Area ABCD = 1
Again let Q (cosα, sinα) & R (cosβ, sin β)
⇒ coordinate of P (cosα– sinα, 0) & S (0,sinβ– cosβ)
PQRS is a square ⇒ PQ ⊥ QR ⇒ slope of QR = -1 = slope of SP
⇒

sin‌β−sin‌α

cos‌β−cosα

=−1=

sin‌β−cos‌β

sin‌α−cos‌α

⇒ sin‌β−sin‌α=−cos‌β+cosα
⇒ sin‌β−cos‌β=sin‌α+cos‌α ………..(i)
and sin‌α+sin‌β=cos‌α+cos‌β ……………..(ii)
⇒ cos‌α=sin‌β
⇒ cos‌α==cos(90−β)
⇒ α+β=90°
Also PQ = QR
⇒ tan‌α=

1

2

Area of PQRS = 2sin2α=2(

1

5

)

Area‌PQRS

Area‌ABCD

=

2∕5

1

=

2

5