The resultant of two rectangular simple harmonic motions of the same frequency and unequal amplitudes but differing in phase by π2 is

If first equation is y1=a1sinω t⇒sinω t=y1a1    ... (i)then second equation will be y2=a2sinω t+π2=a2 sinω tcosπ2+cosω tsinπ2=a2cosω t⇒cosω t=y2a2                                         ... (ii)By squaring and adding equation (i) and (ii)sin2ω t+cos2ω t=y12a12+y22a22  ⇒y12a12+y22a22=1 ; This is the equation of ellipse.