JEE Math Practice Question

If the solution curve $$f(x, y)=0$$ of the differential equation $$\left(1+\log _{e} x\right) \frac{d x}{d y}-x \log _{e} x=e^{y}, x > 0$$, passes through the points $$(1,0)$$ and $$(\alpha, 2)$$, then $$\alpha^{\alpha}$$ is equal to :

1. A.$$e^{\sqrt{2} e^{2}}$$
2. B.$$e^{2 e^{\sqrt{2}}}$$
3. C.$$e^{e^{2}}$$
4. D.$$e^{2 e^{2}}$$