Half-ellipses Intersecting

Geometry Level 5

In the diagram above, points \(A\) through \(E\) are co-linear and evenly spaced. The curves are five half-ellipses, similar to each other, and terminating in the points \(A\) through \(E\).

At the two red points, the half-ellipses intersect at right angles.

At the two green points, the half-ellipses intersect at angle \(0 < \theta < 90^\circ\).

If \(\cos \theta = \dfrac{a}{\sqrt b}\), where \(a\) and \(b\) are coprime positive integers, how much is \(a + b\)?

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