11 x1 t11 06 tangents & normals ii (2012)

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11 x1 t11 06 tangents & normals ii (2012)

  1. 1. Tangents & Normals (ii) Using Cartesian
  2. 2. Tangents & Normals (ii) Using Cartesian(1) Tangent
  3. 3. Tangents & Normals (ii) Using Cartesian(1) Tangent y x 2  4ay x
  4. 4. Tangents & Normals (ii) Using Cartesian(1) Tangent y x 2  4ay P( x1 , y1 ) x
  5. 5. Tangents & Normals (ii) Using Cartesian(1) Tangent y x 2  4ay P( x1 , y1 ) x
  6. 6. Tangents & Normals (ii) Using Cartesian(1) Tangent x2 y x 2  4ay y 4a P( x1 , y1 ) x
  7. 7. Tangents & Normals (ii) Using Cartesian(1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a P( x1 , y1 ) x
  8. 8. Tangents & Normals (ii) Using Cartesian(1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a dy x1 P( x1 , y1 ) when x  x1 ,  dx 2a x
  9. 9. Tangents & Normals (ii) Using Cartesian(1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a dy x1 P( x1 , y1 ) when x  x1 ,  dx 2a x x1  slope of tangent is 2a
  10. 10. Tangents & Normals (ii) Using Cartesian(1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a dy x1 P( x1 , y1 ) when x  x1 ,  dx 2a x x1  slope of tangent is 2a x y  y1  1  x  x1  2a
  11. 11. Tangents & Normals (ii) Using Cartesian(1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a dy x1 P( x1 , y1 ) when x  x1 ,  dx 2a x x1  slope of tangent is 2a x y  y1  1  x  x1  2a 2ay  2ay1  xx1  x12
  12. 12. Tangents & Normals (ii) Using Cartesian(1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a dy x1 P( x1 , y1 ) when x  x1 ,  dx 2a x x1  slope of tangent is 2a x y  y1  1  x  x1  2a 2ay  2ay1  xx1  x12 2ay  2ay1  xx1  4ay1
  13. 13. Tangents & Normals (ii) Using Cartesian(1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a dy x1 P( x1 , y1 ) when x  x1 ,  dx 2a x x1  slope of tangent is 2a x y  y1  1  x  x1  2a 2ay  2ay1  xx1  x12 2ay  2ay1  xx1  4ay1 xx1  2a y  y1 
  14. 14. (2) Normal
  15. 15. (2) Normal y x 2  4ay x
  16. 16. (2) Normal y x 2  4ay P( x1 , y1 ) x
  17. 17. (2) Normal y x 2  4ay P( x1 , y1 ) x
  18. 18. (2) Normal y x 2  4ay P( x1 , y1 ) x1 1 Show the slope of tangent at P is 2a x
  19. 19. (2) Normal y x 2  4ay P( x1 , y1 ) x1 1 Show the slope of tangent at P is 2a 2a x 2  slope of normal is  x1
  20. 20. (2) Normal y x 2  4ay P( x1 , y1 ) x1 1 Show the slope of tangent at P is 2a 2a x 2  slope of normal is  x1  2a y  y1   x  x1  x1
  21. 21. (2) Normal y x 2  4ay P( x1 , y1 ) x1 1 Show the slope of tangent at P is 2a 2a x 2  slope of normal is  x1  2a y  y1   x  x1  x1 x1 y  x1 y1  2ax  2ax1
  22. 22. (2) Normal y x 2  4ay P( x1 , y1 ) x1 1 Show the slope of tangent at P is 2a 2a x 2  slope of normal is  x1  2a y  y1   x  x1  x1 x1 y  x1 y1  2ax  2ax1 2ax  x1 y  2ax1  x1 y1
  23. 23. (3) Line cutting/touching/missing parabola
  24. 24. (3) Line cutting/touching/missing parabola y x 2  4ay x
  25. 25. (3) Line cutting/touching/missing parabola y x 2  4ay x
  26. 26. (3) Line cutting/touching/missing parabola y x 2  4ay x
  27. 27. (3) Line cutting/touching/missing parabola y x 2  4ay y  mx  b x
  28. 28. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x
  29. 29. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x
  30. 30. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 x
  31. 31. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 x
  32. 32. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x
  33. 33. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0
  34. 34. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac
  35. 35. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac   4am   4 1 4ab  2
  36. 36. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac   4am   4 1 4ab  2  16a 2 m 2  16ab  16a  am 2  b 
  37. 37. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac   4am   4 1 4ab  2  16a 2 m 2  16ab  16a  am 2  b   two solutions (cuts) when am2  b  0
  38. 38. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac   4am   4 1 4ab  2  16a 2 m 2  16ab  16a  am 2  b   two solutions (cuts) when am2  b  0 one solution (touches) when am2  b  0
  39. 39. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac   4am   4 1 4ab  2  16a 2 m 2  16ab  16a  am 2  b   two solutions (cuts) when am2  b  0 one solution (touches) when am2  b  0 (common idea)
  40. 40. (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac   4am   4 1 4ab  2  16a 2 m 2  16ab  16a  am 2  b   two solutions (cuts) when am2  b  0 one solution (touches) when am2  b  0 (common idea) no solutions (misses) when am2  b  0
  41. 41. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2).
  42. 42. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b
  43. 43. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b
  44. 44. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m
  45. 45. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m
  46. 46. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y
  47. 47. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m 
  48. 48. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0
  49. 49. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 line is a tangent if   0
  50. 50. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 line is a tangent if   0  4m   4 112m  8  0 2
  51. 51. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 line is a tangent if   0  4m   4 112m  8  0 2 16m2  48m  32  0
  52. 52. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 line is a tangent if   0  4m   4 112m  8  0 2 16m2  48m  32  0 m2  3m  2  0  m  1 m  2   0
  53. 53. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 line is a tangent if   0  4m   4 112m  8  0 2 16m2  48m  32  0 m2  3m  2  0  m  1 m  2   0 m  1 or m  2
  54. 54. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 line is a tangent if   0  4m   4 112m  8  0 2 16m2  48m  32  0 m2  3m  2  0  m  1 m  2   0 m  1 or m  2  tangents are y  x  1 and y  2 x  4
  55. 55. e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 Exercise 9G; 1ac, 2ac, 3a, 4, 7, 9, 11, 12, line is a tangent if   0 13, 15, 17, 18  4m   4 112m  8  0 2 16m2  48m  32  0 m2  3m  2  0  m  1 m  2   0 m  1 or m  2  tangents are y  x  1 and y  2 x  4

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