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

$$\begin{gathered} \mathbf{v}=ky\mathbf{i}+kxj \\ \frac{dx}{dt}=ky,\frac{dy}{dt}=kx \\ \frac{dy}{dx}=\frac{dy}{dt}\times \frac{dt}{dx}=\frac{kx}{ky} \\ y d y =x d x \end{gathered}$$

Integrating and using $\int x^{n}dx=\frac{x^{n+1}}{n+1}$, we have $y^2=x^2+c$