The relation $v^2/(\omega^2A^2) + x^2/A^2 = 1$ represents an ellipse in the $v$-$x$ plane. Semi-axes are $A$ (along $x$) and $A\omega$ (along $v$). This ellipse is a phase-space representation of SHM. The particle traces this ellipse clockwise if $x = A\sin\omega t$. Key check: if a $v$-$x$ graph is given and it is an ellipse centered at origin, the motion is SHM. If it is a circle, then $\omega = 1$ rad/s (a special case).