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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