The slope of the tangent at \((x, y)\) to the curve passing through \((2,1)\) is \(\frac{x^2+y^2}{2 x y}\), then the equation of the curve is

1. A \(a \theta \tan ^2 \theta-a \tan \theta-a \theta+c \quad\left(\right.\) where \(\left.\theta=\tan ^{-1}\left(\sqrt{\frac{x}{a}}\right)\right)\) and c is the constant of integration
2. B \(\mathrm{a} \theta \tan ^2 \theta-\mathrm{a} \tan \theta+\mathrm{a} \theta+\mathrm{c} \quad\left(\right.\) where \(\left.\theta=\tan ^{-1}\left(\sqrt{\frac{x}{\mathrm{a}}}\right)\right)\) and c is the constant of integration
3. C \(a \theta \tan ^2 \theta+a \tan \theta-a \theta+c \quad\left(\text { where } \theta=\tan ^{-1}\left(\sqrt{\frac{x}{a}}\right)\right) \text { and }\)
   c is the constant of integration
4. D \(a \theta \tan ^2 \theta+a \tan \theta+a \theta+c\left(\right.\) where \(\left.\theta=\tan ^{-1}\left(\sqrt{\frac{x}{a}}\right)\right)\) and c is the constant of integration