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A particle is moving with a velocity $$\vec{v} = K(y\hat{i} + x\hat{j})$$, where $$K$$ is a constant. The general equation for its path is:
$$v_x = \frac{dx}{dt} = Ky$$
$$v_y = \frac{dy}{dt} = Kx$$
$$\frac{v_y}{v_x} = \frac{dy/dt}{dx/dt} = \frac{dy}{dx} = \frac{Kx}{Ky} = \frac{x}{y}$$
$$y \, dy = x \, dx$$
$$\int y \, dy = \int x \, dx \implies \frac{y^2}{2} = \frac{x^2}{2} + C'$$
$$y^2 = x^2 + \text{constant}$$
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