Think you know about the Ellipse? Think again!

Geometry Level 5

Consider an ellipse $$E$$ with centre $$C$$. From a point $$K$$, four real, distinct normals are drawn to the ellipse which intersect the major axis of $$E$$ at $$G_1$$, $$G_2$$, $$G_3$$ and $$G_4$$. Let $S=\left (\displaystyle\sum_{i=1}^4CG_i \right)\cdot \left (\displaystyle\sum_{i=1}^4\frac{1}{CG_i} \right)$

Let $$M=\max(S)$$ and $$m=\min(S)$$. Find $$M-m$$

Details and Assumptions:

All lengths are signed.

This problem is part of my set: Geometry

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