Suppose the variation problem $$J[y]=\int_a^bF(x,y,y')dx$$ with free boundary and constraint $\int_a^bG(x,y,y')=l$, how can formulate the corresponding Euler-Lagrange equation?
For fixed boundary i.e. $f(a)=A, f(b)=B$, the Euler-Lagrange equation is $$F_y-\frac{d}{dx}F_{y'}+(G_y-\frac{d}{dx}G_{y'})=0$$
OP's variational problem (v1) is in general ill-posed to begin with.
Example: Let for simplicity
$$G(x,y,y^{\prime}) ~=~y^{\prime} \quad\text{and}\quad F(x,y,y^{\prime})~=~2yy^{\prime} $$
be total derivatives. Then
$$ \ell~=~y(b)-y(a)\quad\text{and}\quad J[y]~=~~y(b)^2-y(a)^2~=~\ell(y(b)+y(a))~=~\ell(\ell +2y(a)), $$
which clearly doesn't have a minimum nor a maximum as there's free boundary conditions.