Problem: A satellite is injected into an orbit at radius r from the center of attraction. The velocity is exactly the desired speed to go into circular orbit but the flight path angle γ≠0. The flight path angle is the angle between the velocity vector and the plane normal to the position vector.

a. Develop the relationship between the resulting eccentricity and sinγ.
b. Develop a relationship between the resulting periapsis distance and sinγ that is valid even for the rectilinear orbit case. Likewise for apoapsis distance.

Here is what I have so far:

I started out with h=rvsin(90-γ). Then substitute h=sqrt(mu*a(1-e^2)). After some algebra, I ended up with:
eccentricity = sqrt(1 - ((r^2*v^2 - r^2*v^2*sinγ^2))/mu/a)

For Part b, I substituted the new eccentricity equation from Part a into the equations for Rp and Ra. Which are Rp = a(1-e) and Ra = a(1+e)

Is what I have done so far correct?

Let m be the mass of the satellite, even though we don't need it. Let a be the semi-major axis. Let e be the eccentricity.

For a circular orbit: v_0^2 = GM/r_0 (*1)

For our setup: v_0^2 = (v_0 sin γ)^2 + (v_0 cos γ)^2

For the extreme points in the orbit, there is no inward or outward motion.
So by the conservation of angular momentum, we must have

v_± = (r_0 v_0 cos γ) / r_± = (√(G M r_0) cos γ) / r_± (*2)

But from conversation of energy we have

m v_0²/ 2 - G M m/r_0 = m v_±² / 2 - G M m/r_±
or
v_±² - v_0² = 2GM (1/r_± - 1/r_0) (*3)

Combining *1, *2 and *3 we get

(G M r_0 cos² γ) / r_±² - GM/r_0 = 2GM (1/r_± - 1/r_0)
or
r_± = r_0 ± r_0 √(sin² γ) (*4)

From r_± = a(1 ± e) we see that we have:

e = | sin γ |