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Chapter 33

Chapter 33. hyperbola. 双曲线. Definition:. The locus of a point P which moves such that the ratio of its distances from a fixed point S and from a fixed straight line ZQ is constant, e , and greater than one .

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Chapter 33

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  1. Chapter 33 hyperbola 双曲线 hyperbola

  2. hyperbola

  3. Definition: The locusof a point P which moves such that the ratio of its distances from a fixed point Sand from a fixed straight line ZQ is constant, e ,and greater than one. S is the focus, ZQ the directrix and the e, eccentricity of the hyperbola. hyperbola

  4. Simplest form of the eqn of a hyperbola e:eccentricity hyperbola

  5. The foci S, S’ are the points (-ae,0), (ae,0) . Q Q’ y The directrices ZQ, Z’Q’ are the lines x=-a/e, x=a/e . B A’ A x O Z’ S Z S’ AA’ is called the transverse axis=2a . 实轴 B’ BB’ is called the conjugate axis=2b . 虚轴 asymptotes hyperbola

  6. e.g. 1 For the hyperbola , find (i) the eccentricity, (ii) the coordinates of the foci (iii) the equations of the directrices and (iv) the equations of the asymptotes . hyperbola

  7. Soln: (i) hyperbola

  8. (ii) Coordinates of the foci are (iii) Equations of directrices are (iv) Equations of asymptotes are i.e. hyperbola

  9. e.g. 2 Find the asymptotes of the hyperbola . Soln: The asymptotes are hyperbola

  10. e.g. 3 Find the equation of hyperbola with focus (1,1); directrix 2x+2y=1; e= . hyperbola

  11. Soln: From definition of a hyperbola, we have PS=ePM . Where PS is the distance from focus to a point P and PM is the distance from the directrix to a point P. e is the eccentricity of the hyperbola. hyperbola

  12. Let P be (x,y). Hence, distance from P to (1,1) is : Distance from P to 2x+2y-1=0 is : hyperbola

  13. is the required equation of the hyperbola. hyperbola

  14. Properties of the hyperbola hyperbola

  15. 1. The curve is symmetrical about both axes. The curve exists for all values of y. 2. The curve does not exist if |x|<a. hyperbola

  16. 3. At the point (a,0) & (-a,0), the gradients are infinite. 4. Asymptotes of the hyperbola : hyperbola

  17. Many results for the hyperbola are obtained from the corresponding results for the ellipse by merely writing in place of . hyperbola

  18. 1. The equation of the tangent to the hyperbola at the point (x’,y’) is 2. The gradient form of the equation of the tangent to the hyperbola is hyperbola

  19. 3. The locus of the midpoints of chords of the hyperbola with gradient m is the diameter : hyperbola

  20. e.g. 4 Show that there are two tangents to the hyperbola parallel to the line y=2x-3 and find their distance apart. hyperbola

  21. Soln: Gradient of tangents=2 Hence, equations of tangents are : Perpendicular distance from (0,0) to the lines are : O Distance= hyperbola

  22. The rectangular hyperbola hyperbola

  23. 1. A hyperbola with perpendicular asymptotes is a rectangular hyperbola. i.e. b=a So, the standard equation of a rectangular hyperbola is : hyperbola

  24. 2. Eccentricity of a rectangular hyperbola = hyperbola

  25. Equation of a rectangular hyperbola with respect to its asymptotes hyperbola

  26. y y x O x hyperbola

  27. Some simple sketches of the rectangular hyperbola : y y xy=-9 xy=9 x o x o y y x x o o hyperbola

  28. hyperbola

  29. Parametric equations of a rectangular hyperbola hyperbola

  30. The equation, is satisfied if t is a parameter. The parametric coordinates of any point are : hyperbola

  31. Tangent and normal at the point (ct,c/t) to the curve Gradient of tangent at (ct,c/t) is hyperbola

  32. Equation of tangent at (ct,c/t) is hyperbola

  33. Equation of normal at (ct,c/t) is hyperbola

  34. e.g. 5 The tangent at any point P on the curve xy=4 meets the asymptotes at Q and R. Show that P is the midpoint of QR. hyperbola

  35. Soln: Let P be the point (2t,2/t). Equation of tangent at P is x-axis and y-axis are the asymptotes. When y=0,Q is (4t,0), when x=0 R is (0,4/t). The midpoint of QR is (2t,2/t). hyperbola

  36. Miscellaneous examples on the hyperbola hyperbola

  37. e.g. 6 A chord RS of the rectangular hyperbola subtends a right angle at a point P on the curve. Prove that RS is parallel to the normal at P. hyperbola

  38. Soln: Let S(ct,c/t), R(cp,c/p) and P(cq,c/q). Gradient of tangent at P is : S R P at hyperbola

  39. Gradient of PS= Gradient of RP= Gradient of RS= Hence, hyperbola

  40. Conclusion: In analytic geometry, the hyperbola is represented by the implicit equation : The condition : B2 − 4AC > 0 • (if A + C = 0, the equation represents a rectangular hyperbola. ) Ellipse

  41. In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. The equation will be of the form : Ax2 + Bxy + Cy2 + Dx + Ey + F = 0with A, B, Cnot all zero. hyperbola

  42. then: • if B2 − 4AC < 0, the equation represents an ellipse (unless the conic is degenerate, for example x2 + y2 + 10 = 0); • ifA = C and B = 0, the equation represents a circle; • if B2 − 4AC = 0, the equation represents a parabola; • if B2 − 4AC > 0, the equation represents a hyperbola; • (if A + C = 0, the equation represents a rectangular hyperbola. ) hyperbola

  43. Analyzing an Hyperbola State the coordinates of the vertices, the coordinates of the foci, the lengths of the transverse and conjugate axes and the equations of the asymptotes of the hyperbola defined by 4x2 - 9y2 + 32x + 18y + 91 = 0. hyperbola

  44. ~ The end ~ hyperbola

  45. Ex 14d do Q1, 3, 5, 7, 9, 11. Misc.14 no need to do hyperbola

  46. Ex 14d Q 1 At point (2a,a/2), Gradient of normal at (2a,a/2) is 4. hyperbola

  47. Equation of normal at (2a,a/2) is : hyperbola

  48. Ex 14d Q 3 (2t,2/t) 2y=x+7 4/t=2t+7 hyperbola

  49. At point A, t=1/2 so, A is (1,4) At point B, t=-4 so, B is (-8,-1/2) x=2t, y=2/t Hence, xy=4 hyperbola

  50. At point A, dy/dx=-4 At point B,dy/dx=-1/16 Eqn of tangent at A : hyperbola

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