Discussion

Explanation:

PA = QB and angles A and B are right angles.

Hence, PABQ is a rectangle.

∴ Required area

= A(rectangle PABQ) – A(segment O-PRQ)

= A(rectangle PABQ) – [A(sector O-PRQ) – A(∆OPQ)]

= A(rectangle PABQ) – A(sector O-PRQ) + A(∆OPQ)

Now, PQ = AB = 25√2 cm and ∠POQ = 90°

∴ Radius of sector = PO = OQ = PQ/√2 = 25 

Now, A(∆OPQ) = (1/2) × OP × OQ
= (1/2) × 25 × 25 = 625/2 sq.cm

A(sector O-PRQ) = (90/360) × π × (OP)²
= (π/4) × (25)² = (625π/4)

Now, height of ∆OPQ = 25/√2

Distance from O to AB is 30 cm

∴ PA = QB = 30 – (25/√2) cm

∴ A(rectangle PABQ) = PA × AB

= [30 – (25/√2)] × (25√2) = 750√2 – 625 sq.cm

∴ Required area = $(750\sqrt{2}-625)$ - $\frac{625\mathrm{\pi }}{4}$ + $\frac{625}{2}$

= 750√2 - 625/2 - 625π/4

= 750√2 - 625$\left(\frac{1}{2}+\frac{\mathrm{\pi }}{4}\right)$

Hence, option (c).

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