Messed Up

Geometry Level pending

In an orthonormal system $$(O; \vec u; \vec v )$$, consider $$M(x,y)$$ moving on $$(E)$$: $$25(x^2+y^2) = (3x-16)^2$$ having a directrix $$(D)$$: $$x = \dfrac{16}3$$. Let $$( \vec u ; \vec{OM} ) = \theta$$. $$(OM)$$ meets $$(E)$$ at $$M'$$.

Given that $$\dfrac1{OM} + \dfrac1{OM'} = \dfrac ab$$, where $$a$$ and $$b$$ are coprime positive integers, find $$a+b$$.

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