Area of a pillow

Geometry Level 5

The graphs of four identical parabolas, as shown above, intersect at the points \(\: (5,5), \:(-5,5), \:(-5,-5) \) and \( (5,-5) \). The vertices of these parabolas are all \(4\) units away from the origin, and the area enclosed by the quadrilateral whose vertices are the foci of these parabolas is \( \frac{1681}{16} \) square units.

If the area formed about the origin enclosed by the four parabolas can be expressed in the form \(\frac{A}{B}\), where \(A\) and \(B\) are coprime positive integers, determine \(A+B\).

Details and Assumptions:

  1. The parabolas are identical in the sense that they have the same focal lengths.

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