homework and exercises - Going downhill - what's the constraint force?

We are going downhill on a path

$$y=a(1-x^2),$$ and I need to calculate the constraint force-position function. What I've done is this:

The lagrangian of the system is

$$L=\frac{1}{2}(\dot{x}^2+\dot{y}^2)-mgy+\lambda(t)(y-a(1-x^2)),$$ so the 2 equation are $$m\ddot{x}=2a\lambda x$$ and $$m\ddot{y}=-mg+\lambda.$$ Based on the equations, the constraint force is $F_y=\lambda$ and $F_x=2a\lambda x$, but I can't calculate the value of $\lambda.$ How should I do it?