If first equation is $y_1 = a_1 \sin \omega t$ ⇒ ⇒ $\dot{\sin \omega t} = \frac{y_1}{a_1}$ ... (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.