Two mutually perpendicular simple harmonic vibrations have same amplitude, frequency and phase. When they superimpose, the resultant form of vibration will be
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(c) If ${y_1} = {a_1}\sin \omega \,t$ and ${y_2} = {a_2}\sin (\omega \,t + 0) = {a_2}\sin \omega \,t$
==> $\frac{{y_1^2}}{{a_1^2}} + \frac{{y_2^2}}{{a_2^2}} - \frac{{2{y_1}{y_2}}}{{{a_1}{a_2}}} = 0$
==> $\frac{{y_1^2}}{{a_1^2}} + \frac{{y_2^2}}{{a_2^2}} - \frac{{2{y_1}{y_2}}}{{{a_1}{a_2}}} = 0$
==> ${y_2} = \frac{{{a_2}}}{{{a_1}}}{y_1}$
This is the equation of straight line.
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