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Question

The maximum area of the rectangle that can be inscribed in a circle of radius 2 units is _____.

A
8πsq.units
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B
4sq.units
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C
5sq.units
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D
8sq.units
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Solution

The correct option is D 8sq.unitsLet the length of the rectangle be h and its breadth be kSince the rectangle is inscribed inside the circle, its diagonal has to be the diameter of the circle.∴h2+k2=42 ...(1)The area of the rectangle is given by hkDifferentiating the condition w.r.t h,2h+2kdkdh=0⇒dkdh=−hk ...(2)Now, differentiating the area since we need to maximize it, k+hdkdh=0 ...(3)Substituting equation (2) into equation (3), we havek+h×−hk=0⇒k2=h2 or k=hSubstituting this relation in equation (1), we get2h2=42 or h2=8⇒ area =hk=h2=8 sq unit

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