Think you know about the Ellipse? Think again!

Geometry Level 4

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 \(M-m\)

Give your answer to 3 decimal places.

Details and Assumptions:

All lengths are signed.

This problem is part of my set: Geometry


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