In this problem, $x_1=3\cos{2t}+4\sin{2t}$ and $x_2=-\sin{2t}+4\cos{2t}$.
$${x_1}^2+{x_2}^2=(3\cos{2t}+4\sin{2t})^2+(-\sin{2t}+4\cos{2t})^2=9(\cos^2{2t}+\sin^2{2t})+16(\cos^2{2t}+\sin^2{2t})=25$$
By definition, this is the equation of a circle centered at the origin with radius 5.
