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Conic Sections

Conic Sections. The Parabola. Introduction. Consider a cone being intersected with a plane. Note the different shaped curves that result. Introduction. They can be described or defined as a set of points which satisfy certain conditions.

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Conic Sections

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  1. Conic Sections The Parabola

  2. Introduction • Consider a cone being intersected with a plane Note the different shaped curves that result

  3. Introduction They can be described or defined as a set of points which satisfy certain conditions • We will consider various conic sections and how they are described analytically • Parabolas • Hyperbolas • Ellipses • Circles

  4. Parabola • Definition • A set of points on the plane that are equidistant from • A fixed line (the directrix) and • A fixed point (the focus) not on the directrix

  5. Parabola • Note the line through the focus, perpendicular to the directrix • Axis of symmetry • Note the point midway between the directrix and the focus • Vertex

  6. Distance = Distance = y + p Equation of Parabola • Let the vertex be at (0, 0) • Axis of symmetry be y-axis • Directrix be the line y = -p (where p > 0) • Focus is then at (0, p) • For any point (x, y) on the parabola

  7. Equation of Parabola • Setting the two distances equal to each other • What happens if p < 0? • What happens if we have . . . simplifying . . . Link to web example

  8. Working with the Formula • Given the equation of a parabola • y = ½ x2 • Determine • The directrix • The focus • Given the focus at (-3,0) and the fact that the vertex is at the origin • Determine the equation

  9. When the Vertex Is (h, k) • Standard form of equation for vertical axis of symmetry • Consider • What are the coordinatesof the focus? • What is the equationof the directrix? (h, k)

  10. When the Vertex Is (h, k) • Standard form of equation for horizontal axis of symmetry • Consider • What are the coordinatesof the focus? • What is the equationof the directrix? (h, k)

  11. Try It Out • Given the equations below, • What is the focus? • What is the directrix?

  12. Another Concept • Given the directrix at x = -1 and focus at (3,2) • Determine the standard form of the parabola

  13. Applications • Reflections of light rays • Parallel raysstrike surfaceof parabola • Reflected backto the focus View Animated Demo How to Find the Focus Build a working parabolic cooker Proof of the Reflection Property

  14. Applications • Light rays leaving the focus reflectout in parallel rays Used for Searchlights Military Searchlights

  15. Assignment • See Handout • Part A 1 – 33 odd • Part B 35 – 43 all

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