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10 – 5 Hyperbolas

10 – 5 Hyperbolas. Hyperbola. Hyperbolas. Has two smooth branches The turning point of each branch is the vertex Transverse Axis: segment connecting the two vertices The two foci lie on the axis of symmetry The center of the hyperbola is the midpoint between the two vertices or the foci

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10 – 5 Hyperbolas

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  1. 10 – 5 Hyperbolas

  2. Hyperbola

  3. Hyperbolas • Has two smooth branches • The turning point of each branch is the vertex • Transverse Axis: segment connecting the two vertices • The two foci lie on the axis of symmetry • The center of the hyperbola is the midpoint between the two vertices or the foci • The Conjugate Axis and the Transverse Axis determine a rectangle that lies between the vertices, and the diagonals of the rectangle determine the asymptotes

  4. Center at (0, 0)

  5. Vertical; Center (0, 0)

  6. Writing and Graphing • A hyperbola centered at (0,0) has vertices (±4, 0) and one focus (5, 0) • What is the standard form equation? • Sketch the Hyperbola • A hyperbola centered at (0,0) has vertices (0, ±4) and one focus (0, 5) • What is the standard form equation? • Sketch the Hyperbola

  7. Analyzing • What are the vertices, foci, and asymptotes of the hyperbola • 9y2 – 7x2 = 63 • 9x2 – 4y2= 36

  8. Reflective Properties • Used in optics • Uses both foci but only one branch reflects • Any ray on the external side of a branch directed at its internal focus will reflect off the branch toward the external focus

  9. Modeling • The graph shows a 2D view of a satellite dish. The focus is located at F1 but the receiving device is located on the bottom of the dish at the point F2. The rays are reflected by the first reflector toward F1 and then reflected by the second reflector toward F2. • What kind of curve is the second reflector? How can you tell? • The vertex of the second reflector is 3 in. from F1 and 21 in. from F2. What is an equation for the second reflector? Assume the conic is horizontal and centered at the origin.

  10. Modeling • Suppose the vertex of the second reflector from the previous problem were 4 in from F1 and 18in from F2. What is the equation for the second reflector? Assume the conic is horizontal and centered at the origin.

  11. Homework • Pg. 650 • # 8 – 30 even, 42 • 13 problems

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