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Question

A rectangle has one side on the positive y-axis and one side on the positive x-axis. The upper right hand vertex of the rectangle lies on the curve y=lnxx2. The maximum area of the rectangle is

A
e1
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B
e1/2
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C
1
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D
e1/2
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Solution

The correct option is A e1
Two sides of triangle are on positive x and y axis.
So, one vertex of the rectangle is the origin.
Let (h,k) be the co-ordinates of the upper right hand vertex of the rectange.
So, the co-ordinates of the other two vertices are (h,0) and (0,k)

(h,k) lies on the curve y=lnxx2
k=lnhh2 ...... (1)

Area of the rectangle = h×k
A=h×lnhh2 ....[Using (1)]
A=lnhh

For area to be maximum, dAdh=0
d(lnhh)dh=0

1lnhh2=0

h=e

A=lnhh=lnee=e1

Hence, the answer is e1.

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