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10.5 Hyperbolas

10.5 Hyperbolas. p.615. 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

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10.5 Hyperbolas

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  1. 10.5 Hyperbolas p.615

  2. 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 • The line thru the foci intersects the hyperbola @ 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

  3. Asymptotes (0,b) Vertex (a,0) Vertex (-a,0) Focus Focus (0,-b) This is an example of a horizontal transverse axis (a, the biggest number, is under the x2 term with the minus before the y)

  4. Vertical transverse axis

  5. Standard Form of Hyperbola w/ center @ origin Foci lie on transverse axis, c units from the center c2 = a2+b2

  6. Graph 4x2 – 9y2 = 36 • Write in standard form (divide through by 36) • a=3 b=2 – because x2 term is ‘+’ transverse axis is horizontal & vertices are (-3,0) & (3,0) • Draw a rectangle centered at the origin. • Draw asymptotes. • Draw hyperbola.

  7. Write the equation of a hyperbole with foci (0,-3) & (0,3) and vertices (0,-2) & (0,2). • Vertical because foci & vertices lie on the y-axis • Center @ origin because f & v are equidistant from the origin • Since c=3 & a=2, c2 = b2 + a2 • 9 = b2 + 4 • 5 = b2 • +/-√5 = b

  8. Assignment

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