A particle executes S.H.M., the graph of velocity as a function of displacement is :

For a body performing SHM, relation between velocity and displacement<br><br>$v = \omega \sqrt {{A^2} - {x^2}}$<br><br>now, square both side<br><br>${v^2} = {w^2}({A^2} - {x^2})$<br><br>$\Rightarrow {v^2} = {w^2}{A^2} - {\omega ^2}{x^2}$<br><br>${v^2} + {\omega ^2}{x^2} = {\omega ^2}{A^2}$<br><br>divide whole equation by ${{\omega ^2}{A^2}}$<br><br>$${{{v^2}} \over {{\omega ^2}{A^2}}} + {{{\omega ^2}{x^2}} \over {{\omega ^2}{A^2}}} = {{{\omega ^2}{x^2}} \over {{\omega ^2}{A^2}}}$$<br><br>${{{v^2}} \over {{{(\omega A)}^2}}} + {{{x^2}} \over {{{(A)}^2}}} = 1$<br><br>above equation is similar as standard equation of ellipses, so graph between velocity and displacement will be ellipses.