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

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What are the coordinates of the foci of the rectangular hyperbola \[x^2-y^2=7 ^2?\]

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What are the coordinates of the vertices of the hyperbola \[4x^2-9y^2-40x+54y -17=0 ?\]

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Given points \(F=(5,0)\) and \(F'=(-5,0),\) what is the equation of the locus of points \(P\) in a plane such that \[ \left\lvert \left\lvert \overline{PF} \right\rvert - \left\lvert \overline{PF'} \right\rvert \right\rvert= 8 ?\]

Note: \(\lvert \overline{AB} \rvert\) denotes the length of line segment \(\overline{AB}.\)

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