Parabolas, ellipses, and hyperbolas, oh my! Learn about this eccentric bunch of shapes.

Find the shortest distance between the parabolas \(2y^2=2x-1\) and \(2x^2=2y-1\).

**Note:** Round off your answer to 2 decimal places.

\(A,B,C,D\) are consecutive vertices of a rectangle whose area is \(2006\) square units. An ellipse with area \(2006\pi\), which passes through \(A\) and \(C\) has its foci at \(B\) and \(D\).

If the perimeter of the rectangle can be expressed as \(a\sqrt{b}\) where \(a\) is a positive integer and \(b\) is a square-free positive integer, find \(a+b\).

Find the maximum area of \(\triangle POQ\).

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