For part a): We can write $$\frac{dy/dt}{dx/dt} = \frac{dy}{dx} = \frac{-8x}{2y}$$

This is a separable equation, so we can write it as

$$2ydy=-8xdx$$

Integrate both sides and get

$$y^2 +c_1 = -4x^2 +c_2$$

rearrange, and let $C = c_2 - c_1$, we get

$$y^2 + 4x^2 = C$$. This is the expression $H(x,y)=C$ that all trajectories of the system satisfies.