A particle which is simultaneously subjected to two perpendicular simple harmonic motions represented by; x = a 1 cos ⁡ ω t and y = a 2 cos ⁡ 2 ω t traces a curve given by :

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x = a 1 cos ⁡ ω t y = a 2 cos ⁡ 2 ω t = a 2 2 cos 2 ⁡ ω t - 1 = a 2 2 x a 1 2 - 1 ∴ y a 2 = 2 x 2 a 1 2 - 1 ∴ a 1 2 y = 2 a 2 x 2 - a 1 2 a 2 which is a parabola symmetrical about y-axis opening in upward direction

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