A particle which is simultaneously subjected to two perpendicular simple harmonic motions represented by; $x = {a_1}\,\cos \,\omega t$ and $y = {a_2}\,\cos \,2\,\omega t$ traces a curve given by
JEE MAIN 2014, Diffcult
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Two perpendicular $S.H.Ms$ are
$x=a_{1} \cos \omega t$ $...(1)$
and $y=a_{2} \cos 2 \omega t$ $...(2)$
From eqn $( 1 )$
$\frac{x}{a_{1}}=\cos w t$
and from eqn $(2)$
$\frac{y}{a_{2}}=\cos 2 \omega t=2 \cos ^{2} \omega t-1$
$y=2 \frac{a_{2}}{a_{1}^{2}} x^{2}-1$
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