==> $\sin \omega \,t = \frac{{{y_1}}}{{{a_1}}}$ ... (i)
then second equation will be ${y_2} = {a_2}\sin \left( {\omega \,t + \frac{\pi }{2}} \right)$
$ = {a_2}\,\left[ {\sin \omega \,t\cos \frac{\pi }{2} + \cos \omega \,t\sin \frac{\pi }{2}} \right] = {a_2}\cos \omega \,t$
==> $\cos \omega \,t = \frac{{{y_2}}}{{{a_2}}}$ ... (ii)
By squaring and adding equation (i) and (ii)
${\sin ^2}\omega \,t + {\cos ^2}\omega \,t = \frac{{y_1^2}}{{a_1^2}} + \frac{{y_2^2}}{{a_2^2}}$
==> $\frac{{y_1^2}}{{a_1^2}} + \frac{{y_2^2}}{{a_2^2}} = 1$; This is the equation of ellipse.