$[$BONUS $(30$pt$)]$ It is well$-$known that the circle minimizes the arc length among all closed curves enclosing a given area. To establish this result, consider the parametric curve $(x(t),y(t)),$ where tin$[0,2),$ representing the curve. The arc length functional is defined as:

$J(x,y)={\displaystyle \underset{0}{\overset{2}{}}}\sqrt[\phantom{2}]{x^{\text{'}}(t{)}^2+y^{\text{'}}(t{)}^2}dt$

with the periodic boundary conditions $x(0)=x(2)$ and $y(0)=y(2).$ Additionally, the curve must satisfy a fixed area constraint:

$W(x,y)=\frac{1}{2}{\displaystyle \underset{0}{\overset{2}{}}}(x(t)y^{\text{'}}(t)-x^{\text{'}}(t)y(t))dt=A.$

Use the results from the previous exercise to prove this statement.