Explanation

(c) If first equation is
$y_{1} = a_{1}\sin{\omega t \Rightarrow \sin{\omega t = \frac{y_{1}}{a_{1}}}}$ …(i)
Then second equation will be
$y_{2} = a_{2}\sin{(\omega t + \frac{\pi}{2})}\ \ $
$= a_{2}\lbrack\sin{\omega t\cos{\frac{\pi}{2} + \cos{\omega t\sin{\frac{\pi}{2}\rbrack = a_{2}\cos{\omega t}}}}}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \cos{\omega t = \frac{y_{2}}{a_{2}}}$ …(ii)
By squaring and adding Eqs. (i) and (ii)
$\sin^{2}{\omega t + \cos^{2}{\omega t = \frac{y_{1}^{2}}{a_{1}^{2}} + \frac{y_{2}^{2}}{a_{2}^{2}}}}$
$\Rightarrow \ \ \ \ \ \ \frac{y_{1}^{2}}{a_{1}^{2}} + \frac{y_{2}^{2}}{a_{2}^{2}} = 1$;
This is equation of an ellipse.