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Question:

All the vertices of a rectangle lie on a circle of radius R. If the perimeter of the rectangle is P, then the area of the rectangle is

Explanation:

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Let the longer side of the rectangle be a and shorter side by b.

In the figure given, (2R)² = a² + b²

⇒ P = 2(a + b)

⇒ $\frac{\mathrm{P}}{2}$ = a + b

Squaring both sides.

⇒ $\frac{{\mathrm{P}}^2}{4}$ = a² + b² + 2ab

⇒ $\frac{{\mathrm{P}}^2}{4}$ - 4R² = 2ab

⇒ $\frac{{\mathrm{P}}^2}{8}$ - 2R² = ab

Area of the rectangle = a × b = ab = $\left(\frac{{\mathrm{P}}^2}{8}-{\mathrm{2R}}^2\right)$

Hence, option (b).