• # question_answer 34) The resultant of two rectangular simple harmonic motions of the same frequency and unequal amplitudes but differing in phase by nil is A)  Simple harmonicB)  CircularC)  EllipticalD)  Parabolic

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 ${{y}_{2}}={{a}_{2}}\sin \left( \omega t+\frac{\pi }{2} \right)$ $={{a}_{2}}\left[ \sin \omega t\cos \frac{\pi }{2}+\cos \omega t\sin \frac{\pi }{2} \right]$ $={{a}_{2}}\cos \omega t$ $\Rightarrow$ $\cos \omega t=\frac{{{y}_{2}}}{{{a}_{2}}}$ ?(ii) By squaring and adding equations (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 the equation of ellipse.