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Conic Sections

Conic Sections. MAT 182 Chapter 11. Four conic sections. Cone intersecting a plane. Hyperbolas Ellipses Parabolas Circles (studied in previous chapter). What you will learn. How to sketch the graph of each conic section.

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Conic Sections

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  1. Conic Sections MAT 182 Chapter 11

  2. Four conic sections Cone intersecting a plane • Hyperbolas • Ellipses • Parabolas • Circles (studied in previous chapter)

  3. What you will learn • How to sketch the graph of each conic section. • How to recognize the equation as a parabola, ellipse, hyperbola, or circle. • How to write the equation for each conic section given the appropriate data.

  4. Definiton of a parabola • A parabola is the set of all points in the plane that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line. • Graph a parabola using this interactive web site. • See notes on parabolas.

  5. Vertical axis of symmetry If x2 = 4 p y the parabola opens UP if p > 0 DOWN if p < 0 Vertex is at (0, 0) Focus is at (0, p) Directrix is y = - p axis of symmetry is x = 0

  6. Translated (vertical axis) (x – h )2 = 4p (y - k) Vertex (h, k) Focus (h, k+p) Directrix y = k - p axis of symmetry x = h

  7. Horizontal Axis of Symmetry If y2 = 4 p x the parabola opens RIGHT if p > 0 LEFT if p < 0 Vertex is at (0, 0) Focus is at (p, 0) Directrix is x = - p axis of symmetry is y = 0

  8. Translated (horizontal axis) (y – k) 2 = 4 p (x – h) Vertex (h, k) Focus (h + p, k) Directrix x = h – p axis of symmetry y = k

  9. Problems - Parabolas • Find the focus, vertex and directrix: 3x + 2y2 + 8y – 4 = 0 • Find the equation in standard form of a parabola with directrix x = -1 and focus (3, 2). • Find the equation in standard form of a parabola with vertex at the origin and focus (5, 0).

  10. Ellipses • Conic section formed when the plane intersects the axis of the cone at angle not 90 degrees. • Definition – set of all points in the plane, the sum of whose distances from two fixed points (foci) is a positive constant. • Graph an ellipse using this interactive web site.

  11. Ellipse center (0, 0) • Major axis - longer axis contains foci • Minor axis - shorter axis • Semi-axis - ½ the length of axis • Center - midpoint of major axis • Vertices - endpoints of the major axis • Foci - two given points on the major axis Focus Center Focus

  12. Equation of Ellipse • a > b • see notes on ellipses

  13. Problems • Graph 4x 2 + 9y2 = 4 • Find the vertices and foci of an ellipse: sketch the graph 4x2 + 9y2 – 8x + 36y + 4 = 0 put in standard form find center, vertices, and foci

  14. Write the equation of the ellipse • Given the center is at (4, -2) the foci are (4, 1) and (4, -5) and the length of the minor axis is 10.

  15. Notes on ellipses • Whispering gallery • Surgery ultrasound - elliptical reflector • Eccentricity of an ellipse e = c/a when e  0 ellipse is more circular when e  1 ellipse is long and thin

  16. Hyperbolas • Definition: set of all points in a plane, the difference between whose distances from two fixed points (foci) is a positive constant. • Differs from an Ellipse whose sum of the distances was a constant.

  17. Parts of hyperbola • Transverse axis (look for the positive sign) • Conjugate axis • Vertices • Foci (will be on the transverse axis) • Center • Asymptotes

  18. Graph a hyperbola • see notes on hyperbolas • Graph • Graph

  19. Put into standard form • 9y2 – 25x2 = 225 • 4x2 –25y2 +16x +50y –109 = 0

  20. Write the equation of hyperbola • Vertices (0, 2) and (0, -2) Foci (0, 3) and (0, -3) • Vertices (-1, 5) and (-1, -1) Foci (-1, 7) and (-1, 3) More Problems

  21. Notes for hyperbola • Eccentricity e = c/a since c > a , e >1 • As the eccentricity gets larger the graph becomes wider and wider • Hyperbolic curves used in navigation to locate ships etc. Use LORAN (Long Range Navigation (using system of transmitters)

  22. Identify the graphs • 4x2 + 9y2-16x - 36y -16 = 0 • 2x2 +3y - 8x + 2 =0 • 5x - 4y2 - 24 -11=0 • 9x2 - 25y2 - 18x +50y = 0 • 2x2 + 2y2 = 10 • (x+1)2 + (y- 4) 2 = (x + 3)2

  23. Match Conics • Click here for a matching conic section worksheet.

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