Planet moves in an elliptical orbit around one of the foci. The ratio of maximum velocity Vmax and minimum velocity Vmin ansd eccentricity e of the ellipse is given by
from the kepler's law of orbit.
Ra=a(1+e)Rp=a(1−e)
From the conservation of angular momentum
mvaRa=mvpRp
vmaxvmin=vpva=RaRp=1+e1−e
Hence, the ratio is 1+e1−e