A) Simple harmonic
B) Circular
C) Elliptical
D) Parabolic
Correct Answer: C
Solution :
If first equation is \[{{y}_{1}}={{a}_{1}}\sin \omega \,t\]Þ \[\sin \omega \,t=\frac{{{y}_{1}}}{{{a}_{1}}}\] ... (i) then second equation will be \[{{y}_{2}}={{a}_{2}}\sin \left( \omega \,t+\frac{\pi }{2} \right)\] \[={{a}_{2}}\,\left[ \sin \omega \,t\cos \frac{\pi }{2}+\cos \omega \,t\sin \frac{\pi }{2} \right]={{a}_{2}}\cos \omega \,t\] Þ \[\cos \omega \,t=\frac{{{y}_{2}}}{{{a}_{2}}}\] ... (ii) By squaring and adding equation (i) and (ii) \[{{\sin }^{2}}\omega \,t+{{\cos }^{2}}\omega \,t=\frac{y_{1}^{2}}{a_{1}^{2}}+\frac{y_{2}^{2}}{a_{2}^{2}}\] Þ \[\frac{y_{1}^{2}}{a_{1}^{2}}+\frac{y_{2}^{2}}{a_{2}^{2}}=1\]; This is the equation of ellipse.
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