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Question

The area of the figure bounded by a curve, the x-axis, and two ordinates, one of which is constant the other variable, is equal to ratio of the cube of the variable ordinate to the variable abscissa. The curve is given by

A
(2y2x2)2=Cx2
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B
(2y2x2)3=Cx2
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C
(y2x2)3=Cx2
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D
(2yx2)3=Cx2
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Solution

The correct option is A (2y2x2)3=Cx2
Area of the shaded region haf(x)dx=f3(h)h (given)
Differentiating both sides w.r.t 'h', we get
f(h)=h3t2(h).f(h)f2(h)h2h2f(h)=3hf2(h)f(h)f3(h) ...(1)
Replace f(h) by y and h by x
x2y=3xyy2dydxy3dydx=x2y+y33xy2 ...(2)
Equation (2) is a homogeneous differential equation
Substitute y=vxdydx=v+xdvdx in (2)

v+xdvdx=v+v33v23v2dv2v3v=dxx

3v2v3v=dxx344vdv2v21=dxx

34log(2v21)=log(cx)(2v21)34=cx

v=yx(2y2x2)3=cx2

357277_209168_ans_aeba8f05fc0847299ac896314e4f0a8a.png

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