Solution to the Dipole - Field line problem

 Its obvious enough, that for a single charge, number of field lines in a solid angle is constant and proportional to the first power of the magnitude of the charge. (as hinted by the gauss' law)
(If you disagree.. click here.)

Now, for the dipole case we have at hand, this is valid if we take the section of a sphere of infinitesimally small radius where the other charge's effect is negligible. But that, doesn't matter. It simply means we can continue.
Further, we know, that a solid angle in a cone with semi-vertical angle ß is proportional to sin^2 (ß/2) (using the definition of a solid angle as in the link above).

In short, no. of lines through a circular patch somewhere near an isolated charge, is proportional to the magnitude of the charge and sin^2 (ß/2) where ß is the semi-vertical angle of the subtended cone.

Now for the problem here, consider two infinitesimally small cones around the 2 charges (aligned along the line connecting them), making semi-vert. angles α and ß, such that, the number of field lines through both of them are EQUAL. So, α & ß are actually the angles made by a particular field line at the 2 charges!

Hence the answer:                Q.(1-cosα) = q.(1-cosß)

Comments invited.
:)