A particle is moving with a velocity v→=kyi^+xj^ where k is constant. The general equation for its path is

v→=kyi^+xj^ ⇒vxi+vyj=kyi^+xj^ ⇒dxdti+dydtj=kyi^+xj^ comparing, ⇒dxdt=ky;   dydt=kx divide one by the other above equations dxdtdydt=kykx ⇒dxdy=yx ⇒xdx=ydy integrating ∫xdx=∫ydy ⇒y2=x2+c