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

The figure above shows two ellipses whose major axes are perpendicular to each other. Each ellipse passes through the other ellipse’s foci, which form the vertices of a square. If the shaded square encloses an area of 16, then what is the area enclosed by one of the ellipses?

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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.

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\(A,B,C,D\) 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 \(a+b\).

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Consider an ellipse \(\dfrac{x^2}{144} + \dfrac{y^2}{64} = 1\). A line is drawn tangent to the ellipse at a point \(P\). A line segment drawn from the origin to a point \(Q\) on this line is perpendicular to this tangent line.

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

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