A particle is moving with a 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 :
From given equation,
$$\eqalign{ & \vec v = k\left( {y\hat i + x\hat j} \right) = ky\hat i + kx\hat j = {V_x}\hat i + {V_y}\hat j \cr & \frac{{dx}}{{dt}} = ky\,\,{\text{and}}\,\,\frac{{dy}}{{dt}} = kx \cr & {\text{Now}},\,\,\frac{{\frac{{dy}}{{dt}}}}{{\frac{{dx}}{{dt}}}} = \frac{x}{y} = \frac{{dy}}{{dx}}\,\,\, \Rightarrow ydy = xdx \cr} $$
Integrating both sides we get $${y^2} = {x^2} +$$   constant