If a curve \( y=f(\mathrm{x}) \), passing through the point \( (1,2...

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If a curve \( y=f(\mathrm{x}) \), passing through the point \( (1,2) \), is the solution of the differential equation,
\( \mathrm{P} \) \( 2 x^{2} d y=\left(2 x y+y^{2}\right) d x \), then \( f\left(\frac{1}{2}\right) \) is equal to :

W
[JEE Main-2020 (September)]
(a) \( \frac{1}{1+\log _{e} 2} \)
(b) \( \frac{-1}{1+\log _{\mathrm{e}} 2} \)
(c) \( 1+\log _{e} 2 \)
(d) \( \frac{1}{1-\log _{e} 2} \)
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