Semi-semicircle

Geometry Level 4

Take a semicircle of radius $$R$$ and one semicircle $$k$$ times bigger and make them tangent like in the picture above. Then construct a $${{90}^{\circ }}$$ arc through the midpoints of these 2 semicircles. If the shaded area is $$A$$, calculate the ratio $$\dfrac{A}{kR^2}$$.

Hint: Notice that all three centers are collinear.

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