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Understanding Conic Sections: Definitions, Properties, and Equations

This chapter delves into conic sections, exploring their definitions, properties, and equations. A conic is formed by the intersection of a plane and a double-napped cone, resulting in shapes such as circles, ellipses, parabolas, and hyperbolas. We discuss degenerate conics, the algebraic definitions of each conic type, and the standard equations. Key objectives include recognizing, graphing, and writing equations for conics centered at the origin, and finding focal points. Examples demonstrate how to determine the standard form from general equations.

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Understanding Conic Sections: Definitions, Properties, and Equations

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  1. Mathematics 116Bittinger • Chapter 7 • Conics

  2. Mathematics 116 • Conics • A conic is the intersection of a plane an a double-napped cone.

  3. Degenerate Conic • Degenerate conic – plane passes through the vertex • Point • Line • Two intersecting lines

  4. Algebraic Definition of Conic

  5. Definition of Conic • Locus (collection) of points satisfying a certain geometric property.

  6. Circle • A circle is the set of all points (x,y) that are equidistant from a fixed point (h,k) • The fixed point is the center. • The fixed distance is the radius

  7. Algebraic def of Circle • Center is (h,k) • Radius is r

  8. Equation of circlewith center at origin

  9. Def: Parabola • A parabola is the set of all points (x,y) that are equidistant from a fixed line, the directrix, and a fixed point, the focus, not on the line.

  10. Standard Equation of ParabolaVertex at Origin • Vertex at (0,0) • Directrix y = -p • Vertical axis of symmetry

  11. Standard Equation of ParabolaOpening left and right • Vertex: (0,0O • Directrix: x = -p • Axis of symmetry is horizontal

  12. Willa Cather – U.S. novelist (1873-1947) • “The higher processes are all simplification.”

  13. Definition: Ellipse • An ellipse is the set of all points (x,y), the sum of whose distances from two distinct points (foci) is a constant.

  14. Standard Equation of EllipseCenter at Origin • Major or focal axis is horizontal

  15. Standard Equation of EllipseCenter at Origin • Focal axis is vertical

  16. Ellipse: Finding a or b or c

  17. Definition: hyperbola • A hyperbola is the set of all points (x,y) in a plane, the difference whose distances from two distinct fixed points (foci) is a positive constant.

  18. Hyperbola equationopening left and rightcentered at origin

  19. Standard Equation of Hyperbolaopening up and downcentered at origin

  20. Hyperbolafinding a or b or c

  21. Objective – Conics centered at origin • Recognize, graph and write equations of • Circle • Parabola • Ellipse • Hyperbola • Find focal points

  22. Rose Hoffman – elementary schoolteacher • “Discipline is the keynote to learning. Discipline has been the great factor in my life.”

  23. Mathematics 116 • Translations • Of • Conics

  24. Circle • Center at (h,k) radius = r

  25. Ellipse major axis horizontal

  26. Ellipsemajor axis vertical

  27. Hyperbolaopening left and right

  28. Hyperbolaopening up and down

  29. Parabolavertex (h,k) opening up and down

  30. Parabola vertex (h,k)opening left and right

  31. Objective • Recognize equations of conics that have been shifted vertically and/or horizontally in the plane.

  32. Objective • Find the standard form of a conic – circle, parabola, ellipse, or hyperbola given general algebraic equation.

  33. Example • Determine standard form – sketch • Find domain, range, focal points

  34. Example - problem • Determine standard form – sketch • Find domain, range, focal points

  35. Winston Churchill • “It’s not enough that we do our best; sometimes we have to do what’s required.”

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