A particle is moving with velocity $$\vec v = k\left( {y\,\hat i + x\,\hat j} \right),$$    where $$k$$  is a constant. The general equation for its path is-

Solution :
$$\eqalign{ & \vec v = k\left( {y\,\hat i + x\,\hat j} \right) = {v_x}\hat i + {v_y}\hat j = \frac{{dx}}{{dt}}\hat i + \frac{{dy}}{{dt}}\hat j \cr & \therefore \frac{{dx}}{{dt}} = ky\,\,{\text{and }}\therefore \frac{{dy}}{{dt}} = kx \cr & \therefore \frac{{dy}}{{dx}} = \frac{x}{y} \cr & \Rightarrow y\,dy = x\,dx \cr & \Rightarrow {y^2} = {x^2} + \,\,{\text{constant}} \cr} $$