Two planets A and B move around the Sun in elliptic orbits with time periods $T_A$ and $T_B$ respectively. If the eccentricity of the orbit of B is ε and its distance of closest approach to the Sun is R, then the maximum possible distance between the planets is?
Attempt at a solution
I think i am almost there but this is what i have so far. Using the eccentricity and R, I found the expression for the semi major axis of B as $R_{SB}$=$\frac{R}{1-\epsilon}$ and using the relation for time periods, i have the expression for $R_{SA}$=$[\frac{T_A}{T_B}]^{2/3}$$\frac{R}{1-\epsilon}$ Now my doubt is whether to assume that two planets have sun at the same focus(say, on the right side ...picture concentric ellipses) in which case the maximum distance is when one of them is closest to the sun and the other is farthest OR if the two planets have sun at different focii (one to the left and other to the right, in this the orbits overlap partly; in either of the case, i cannot find a perfect expression for the maximum distance!
These are the two cases i am picturing, of course there might be many more oriented differently in the 2D. In the diagram on the left, max distance is when both are at their apogee. In the diagram on the left, its when one is at the apogee and the other is at the perigee. Is it possible to find an expression for distance which i can differentiate to find the max value?
Answer is $\frac{1+\epsilon}{1-\epsilon}$(1+($\frac{T_A}{T_B})^\frac{2}{3})$R