Solve $I[y]=\int_{x_0}^{x_1}y^{-\frac{1}{2}}(1+(y')^2)^\frac{1}{2} \mathrm dx$ parametrically

Question

If $$I[y]=\int_{x_0}^{x_1}F(x,y,y') \mathrm dx$$ Where $$F=y^{-\frac{1}{2}}(1+(y')^2)^\frac{1}{2}$$ Then I have shown the Euler-Lagrange equation implies that $$y(1+(y')^2)=2a$$ For some $a\geq0$. If originally $x=y=0$ can anyone help me show that the parametric solution to this problem is the following? $$x=a(t-\operatorname{sin}t)$$ $$y=a(1-\operatorname{cos}t)$$

Answer 1

$$x=a(t-\operatorname{sin}t)$$ $$y=a(1-\operatorname{cos}t)$$ Implies $$\dot{x}=a(1-\operatorname{cos}t)$$ $$\dot{y}=a\operatorname{sin}t$$ Then $$y'=\frac{\operatorname{sin}t}{1-\operatorname{cos}t}$$ Therefore $$y(1+(y')^2)=\frac{a(1-2\operatorname{cos}t+1)}{1-\operatorname{cos}t}=2a$$ Therefore it parametrically describes the solution.