A) Simple harmonic
B) Circular
C) Elliptical
D) Parabolic
Correct Answer: C
Solution :
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.
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