Another Ellipse Problem

Geometry Level 4

A variable point \(P\) on an ellipse of eccentricity \(e=\dfrac{1}{8}\), is joined to it's focii \( S_1 \) and \(S_2 \).

Given that the locus of the incentre of the triangle \( \Delta P S_1 S_2 \) comes out to be a conic;

Evaluate its eccentricity \(e'\). Now \(e' \) is of the form \(\frac{\sqrt{a}}{b}\), where \(a\) and \(b\) are coprime positive integers, find \(a\times b\).

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