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

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The above diagram is an ellipse-shaped track. The distance between the two foci \(F\) and \(F'\) is \(8,\) and the sum of the distances between any point \(P\) on the inner side of the track and \(F\) and \(F'\) is \(32.\) What is the maximum area of \(\triangle PF'F?\)

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For an ellipse \(E\) with foci \(A=(2,0)\) and \(B=(10,0),\) every point \(P\) on \(E\) satisfies \( \lvert PA \rvert + \lvert PB \rvert = 30.\) What is the equation of ellipse \(E?\)

Note: \( \lvert PA \rvert\) denotes the length of line segment \(PA\).

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In the above diagram, \(F\) and \(F'\) are the foci of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) and \(A\) and \(A'\) are the end points of the major axis of the ellipse. If the area of triangle \(\triangle A'PA\) is double the area of triangle \(\triangle F'PF\) and the perimeter of triangle \(\triangle F'PF\) is \(66,\) what is the value of \(a^2+b^2?\)

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