## Semimajor Axis and Eccentricity after increased velocity

Question:

A space shuttle is in an orbit about the Earth. At its apogee, it uses thrusters and increases its velocity by 400 m/sec. What is the new orbit semimajor axis, eccentricity and how much will the next perigee altitude be increased?

Known:

Original semimajor axis: 7000 km -> a
Original eccentricity: 0.05 -> e
u= GxEarth's Mass=3.986x10$$^{5}$$

What I have done so far:

I figured out the apogee and perigee of the orbit, as well as the velocity at the apogee before the firing of the thrusters.

i) apogee: a(1+e) = 7350 km
ii) perigee: a(1-e) = 6650 km
iii) velocity at apogee:

$$\sqrt{u*((2/r)-(1/a))}/$$ where r = apogee.

I got v=7.17 km/s

Now after the thrusters are fired, the new velocity is 7.57 km/s

Using $$\epsilon$$ = V$$^{2}$$$$/2$$ - u$$/r$$ where r is the current position, aka the apogee and plugging $$\epsilon$$ into

a = -u$$/2\epsilon$$

I found the new semimajor axis to be 7809 km. But then here is the problem. How do I find out the new eccentricity? Thanks!
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