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Question

Given that the slope of the tangent to a curve y=y(x) at any point (x,y) is 2yx2. If the curve passes through the centre of the circle x2+y22x2y=0, then its equation is :

A
xloge|y|=2(x1)
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B
xloge|y|=(x1)
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C
xloge|y|=2(x1)
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D
x2loge|y|=2(x1)
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Solution

The correct option is A xloge|y|=2(x1)
dydx=2yx2
dyy=2dxx2
Integrating both sides
dyy=2dxx2
loge|y|=2x+c

The curve passes through the centre of the circle x2+y22x2y=0, i.e., (1,1)
0=2+c
c=2
Equation of the curve is
loge|y|=2x+2
xloge|y|=2(x1)

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