When a particle executes simple Harmonic motion, the nature of...

The displacement equation of a particle is given by

$x = Asin\left(\omegat + ϕ\right) ⇒ \sin \left(\omegat + ϕ\right) = \dfrac{x}{A}$

The velocity equation is given by,

$v = \omega Acos\left(\omegat + ϕ\right) ⇒ \cos \left(\omegat + ϕ\right) = \dfrac{v}{\omegaA}$

Since, $\sin^{2}\theta + \cos^{2}\theta = 1$

$⇒ \dfrac{{x}^{2}}{{A}^{2}} + \dfrac{{v}^{2}}{{\omega}^{2}{A}^{2}} = 1$

The above equation represents an ellipse.