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10.5 Hyperbolas What you should learn: Goal1 Goal2 Graph and write equations of Hyperbolas. Identify the Vertices and Foci of the hyperbola. 10.5 Hyperbolas Goal3 Identify the Foci and Asymptotes.

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Hyperbolas Like an ellipse but instead of the sum of distances it is the difference A hyperbola is the set of all points P such that the differences from P to two fixed points, called foci, is constant 10.5 Hyperbolas

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Hyperbolas The line thru the foci intersects the hyperbola at two points (the vertices) The line segment joining the vertices is the transverse axis, and it’s midpoint is the center of the hyperbola. Has 2 branches and 2 asymptotes The asymptotes contain the diagonals of a rectangle centered at the hyperbolas center 10.5 Hyperbolas

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Standard Form of Hyperbola w/ center at origin Equation Transverse Axis AsymptotesVertices Horizontal y =+/- ( b/a ) x (+/- a,0) Vertical y =+/- ( a/b ) x (0,+/- a ) Foci lie on transverse axis, c units from the center c 2 = a 2 +b 2 10.5 Hyperbolas

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(0, b ) (0,-b ) Vertex ( a,0 ) Vertex (-a,0 ) Asymptotes This is an example of a Horizontal Transverse axis (a, the biggest number, is under the x 2 term with the minus before the y) Focus 10.5 Hyperbolas

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Vertical Transverse axis 10.5 Hyperbolas

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36 a = 3 b = 2 because term is positive, the transverse axis is horizontal & vertices are (-3,0) & (3,0) Example) Graph the equation. 10.5 Hyperbolas

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Graph 4x 2 – 9y 2 = 36 Draw a rectangle centered at the origin. Draw asymptotes. Draw hyperbola. Example) 10.5 Hyperbolas

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Write the equation of a hyperbola with foci (0,-3) & (0,3) and vertices (0,-2) & (0,2). Transverse axis is Vertical because foci & vertices lie on the y-axis Center is the origin because foci & vertices are equidistant from the origin Since c = 3 & a = 2, c 2 = b 2 + a 2 9 = b 2 + 4 5 = b 2 +/-√5 = b 10.5 Hyperbolas

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Reflection on the Section How are the definitions of ellipse and hyperbola alike? How are they different? assignment Both involve all points a certain distance from 2 foci; For an ellipse, the sum of the distances is constant; for a hyperbola, the difference is constant. 10.5 Hyperbolas Page 618 # 15 – 63 odd

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