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Question

# Rectangle of maximum area that can be inscribed in an equilateral triangle of side a will have area =

A
a232
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B
a234
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C
a238
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D
none
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Solution

## The correct option is C a2√38Let the side BC=a be chosen along x-axis and altitude AD be along y-axis. AD2=AC2−DC2=a2−a24=3a24 ∴AD=a√32 Let QPSR be the rectangle inscribed in the triangle. If A be its area, then A=2xy where (x,y) are the co-ordinates of vertex P which lies on line AC whose equation in intercept form is xa/2+y√3a/2=1 or 2xa+2ya√3=1⋯(1) Area A=2xy=x(1−2xa)a√3 ...[ from (1) ]dAdx=a√3(1−4xa)=0 ∴xa4 d2Adx2=− ive and A is maximum.∴A=a4(1−12)a√3=a2√38Ans: C

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