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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 a238
Let the side BC=a be chosen along x-axis and altitude AD be along y-axis.
AD2=AC2DC2=a2a24=3a24 AD=a32
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+y3a/2=1 or 2xa+2ya3=1(1)
Area A=2xy=x(12xa)a3 ...[ from (1) ]
dAdx=a3(14xa)=0
xa4 d2Adx2= ive and A is maximum.
A=a4(112)a3=a238
Ans: C

214586_184502_ans.bmp

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