An object is moving in an ellipse of eccentricity e under the action of a force F towards a force. When the object is at the periapsis, F is suddenly transferred to the other focus. Show that eccentricity of the new orbit is $\frac{e(3+e)}{(1-e)}$.
Approach
Right as the object comes to the periapsis, the energy equation is:
$\varepsilon_1 = \frac{V^2 _p}{2} + \frac{\mu}{r_p} = -\frac{\mu}{2a} $.
I am assuming that the energy will be conserved before and after the change in the direction of the force.
$\varepsilon_2 = \frac{V^2 _2}{2} + \frac{\mu}{r_2}$.
Other than this, I am not sure what I am supposed to do.
Any help would be appreciated.