c
(c) If first equation is \({y_1} = {a_1}\sin \omega \,t\)

==> \(\sin \omega \,t = \frac{{{y_1}}}{{{a_1}}}\) ... (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.