Deriving Mechanics Equations for the Moon

[Dime]

Alright, so I have a few questions.
a) Consider a moon of mass m orbiting a planet of mass M in an elliptical orbit. The equation for the radius of an elliptical orbit is r=a(1-e²) / (1+ecos$$\vartheta$$). Derive equations for the orbital radius of the moon when it closest to and farthest from the planet.
My answer turns out to be periapse: r=a(1-e)
and for apoapse: r=a(1+e) a is the semi major axis of the ellipse and e is the eccentricity

b) Combing your result from a with angular momentum (L=mvr), derive an equations for the ratio of the moon's speed at periapse and apoapse in terms of eccentricty, that is v_p/v_a.

I ended up getting v_p/v_a=(1+e) / (1-e)​

c) Write down the equation which equates the total energy of the moon at apoapse with its total energy at periapse using the equation E_p=-GMm / r
I am getting confused with this one. So the total energy would be the gravitational potential energy plus the kinetic energy right? So for the energy at periapse would look like

E=(-GMm / r) + 1/2mv²​

the velocity being the one derived earlier.
then using the equations derived in the first couple parts I end up getting

E=(-2GMm + mv(1-e)) / (2a(1-e)​

And somehow I think that is wrong. The next part asks me to show that at periapse

v²=(GM / a)[(1+e)/(1-e)]​

but I may be over complicating things for part c and trying to solve d at the same time I am not sure. Anyone have any ideas if I made a mistake earlier or if they can offer some advice in solving it?

 

[Dime]

nevermind I figured it out