Euler Lagrange to solve for maximum value of F if the end points of f(x) are varying

Question

Maximum value of $F(x,f'(x))$ which is given by $$F(x,f'(x))=\int_z^{a+z} 2x\sqrt{1+f'(x)^2}dx $$ ($z$ is lying on the line $y=2x$), $f(x)$ is symmetric with respect to $x$ axis. Such that $$ \int_z^{z+a} \sqrt{1+f'(x)^2} dx$$ is constant. What should the Euler-Lagrange for this setup look like so as to maximimize value of $f(x)$. As such what I have done in past years was that the a point was fixed but here its varying. What would the equation looks like? Basically what I am asking was if the end points are varying what is the equation turn out to be which one needs to solve considering curve length is constant? Turns out shape is catenary can anyone show it?