When a satellite in a circular orbit around a planet is given a sudden speed boost, it tends to settle into an elliptical orbit, with the planet at one focus. Why does it settle into an elliptical orbit rather than a circular one with the corresponding, appropriate radius? Also, why does the planet have to be at one focus of the ellipse?
I have worked through the calculations, and I understand that when the speed boost is given, the satellite is moving too fast to remain in the current circular orbit, and hence deviates from the circle. What I want to understand is why doesn't its deviation lead it to relocate to another circular orbit further away from the planet? Also, moving to a higher orbit requires positive work, which is done by whatever provided the speed boost, so the satellite should be able to join a higher circular orbit.