Why does a eutectic point exist for a system of alloy ? Why is it that there is such a definite ratio of composition at which this occurs ?
First, there are several other answers that you might be interested in looking at first (full disclosure - they include some by me).
These together go in to some detail on why the eutectic forms (when the liquid free energy drops down and touches the tie-line between the two solid phases - the Physics one), and the interactions one needs to get the liquid phase to drop below the tie line between the solids (the Chemistry one). To get a eutectic you need an interaction term in the liquid Gibbs free energy.
As for why it happens at such a definite composition, the glib answer is the Gibbs phase rule. In a system at fixed pressure, three phases can coexist at only one point.
Given that this question is, in general, a duplicate of the ones listed above, particularly of the Chemistry SE one, I have no particular need of the bounty. Go read those and see if they help.