Orbital energy of ellipse and hyperbolic trajectory

[IPhO' 2008]

Please tell me how to find the orbital energy of ellipse and hyperbolic trajectory.
Thank you.

 

[Janus]

In both cases, it is the sum of the kinetic energy and gravitational potential at any point of the trajectory.

Alternatively, if you know the semi-major axis(a), it can be found by:

$$-\frac{GM}{2a}$$
for an ellipse

and

$$\frac{GM}{2a}$$ for a hyperbola.

 

[IPhO' 2008]

Please show me how to prove it.