Distribution of $Y_1$ and $Y_2$

$$X_1$$ and $$X_2$$ be two independent Normal $$(0,\sigma^2)$$ random variables. Define $$Y_1={\sqrt{X_1^2 +X_2^2}}$$ and $$Y_2=\frac{X_1}{\sqrt{Y_1}}$$ . Show that $$Y_1$$ and $$Y_2$$ are independently distributed. I tried to use transformation technique $$(X_1,X_2) \to (Y_1,Y_2)$$ but as the mapping is not bijective, I got confused. Please help me out.