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Conic Sections
Parabolas, ellipses, and hyperbolas, oh my! Learn about this eccentric bunch of shapes.
Excel in math and science
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Consider a branch of the hyperbola \[x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-3=0\] with vertex at a point \(A.\) Also, let \(B\) be one of the end points of its latus rectum. If \(C\) is the focus of the hyperbola nearest to point \(A,\) what is the area of triangle \(ABC ?\)
Excel in math and science
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by Brilliant Staff
What are the coordinates of the foci of the rectangular hyperbola \[x^2-y^2=7 ^2?\]
Excel in math and science
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What are the coordinates of the vertices of the hyperbola \[4x^2-9y^2-40x+54y -17=0 ?\]
Excel in math and science
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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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Suppose that \((7,12)\) and \((11,8)\) are the foci of a hyperbola passing through the origin. Then the eccentricity of the hyperbola is \(\text{_________}.\)
Excel in math and science
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by Brilliant Staff