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10.1 & 10.2

10.1 & 10.2. CONICS. Why do we need to learn conics?. Conics have been used for hundred of years to model and solve engineer problems.

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10.1 & 10.2

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  1. 10.1 & 10.2 CONICS

  2. Why do we need to learn conics? • Conics have been used for hundred of years to model and solve engineer problems. • For example, parabola can be used to model the cross section of a television dish antenna. Ellipse can be used to model the equation of a satellite’s orbit around Earth.

  3. Conic sections were discovered during 600 to 300 B.C. (the classical Greek period) • A conic section (or conic) is the intersection of a plane and a double-napped cone

  4. Basic Conics

  5. Degenerate Conics

  6. Circle • Definition of a circle: A collection of all points (x,y) that are equidistant from a fixed point (h,k) • Equations of a circle at the origin (0,0) x2 + y2 = r2 (r is the radius)

  7. Ex) Find the equation of a circle that has its vertex at the origin and has the diameter d = 8 Equation of a circle:x2 + y2 = r2 x2 + y2 = 42 (d=8 so r=4) x2 + y2 = 16

  8. Parabola • Definition of a parabola: A set of all points (x,y) that are equidistant from a fixed line, the directrix, and a fixed point, the focus. The midpoint between the focus and the directrix is the vertex of the parabola and the line passing through the focus and the vertex is the axis of the parabola.

  9. EQUATION: y2 = 4px EQUATION: x2 = 4py

  10. Ex) Write the standard form of the equation of the parabola with vertex at the origin and focus at (2,0) Equation: y2 = 4px p = 2 so y2 = 4(2) x y2 = 8 x

  11. Ellipse • Definition of an ellipse: • A set of points (x,y) in the plane, the sum of whose distances from 2 distinct fixed points (foci) is a contant.

  12. GIVEN: c2 = a2 – b2

  13. Ex) Find the equation of an ellipse that has foci (-2, 0) and (2, 0) and the vertex on the major axis is (-3,0) and (3,0) We have a = 3 and c = 2 Also: c2 = a2 – b2 22 = 32 – b2 5 = b2 so b = 5

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

  15. GIVEN: c2 - a2 = b2

  16. Ex: Find the standard form of the equation of a hyperbola with foci (-3,0) and (3,0) and vertices (-2,0) and (2,0) We have c = 3 and b = 2 b2 = c2 –a2 22 = 32 –a2 a = 5

  17. Identify conics: • 5y2 – 25x2 – 75 • 7x2 + y2 – 49 = 0

  18. 2) Given parabola: x2 +12y = 0 and the tangent line x + y - 2 = 0. Graph using your graphing calculator and find the point of the intercept.

  19. 3) Given parabola: y2 – 8x = 0 and the tangent line x – y + 2 = 0. Graph using your graphing calculator and find the point of the intercept.

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