we can consider eccentricity to be a general term represented by e. Apogee is the point in the elliptical orbit where the planet is farthest from the sun and perigee is the point where the planet is nearest to sun.In order to calculate the ratio of kinetic energy, we can use conservation of angular momentum.Now, we know that angular momentum is conserved for the planet throughout the orbit. Hence,Angular momentum at apogee = angular momentum at perigeeLet rabe the distance at apogee and rpbe the distance at perigee. The value of r1andr2can be represented in terms of the semi major axis and eccentricity (e) as mentioned below. If a is the length of semi major axis, thenra= a(1+e)and,rp= a(1-e)Now, by conservation of angular momentum, if the mass of the planet is m and the velocity at apogee and perigee be vaand vprespectively.m vara= m vprp, orva/vp=rp/ra= (1-e)/(1+e)The ratio of kinetic energy is given by = (1/2mvp2)/(1/2mva2) = (vp/va)2=[(1+e)/(1-e)]2