11 x1 t11 06 tangents & normals ii (2012)

Download 11 x1 t11 06 tangents & normals ii (2012)

Post on 25-May-2015

878 views

Category:

Education

2 download

Embed Size (px)

TRANSCRIPT

<ul><li> 1. Tangents &amp; Normals (ii) Using Cartesian</li></ul> <p> 2. Tangents &amp; Normals(ii) Using Cartesian(1) Tangent 3. Tangents &amp; Normals(ii) Using Cartesian(1) Tangent yx 2 4ay x 4. Tangents &amp; Normals (ii) Using Cartesian(1) Tangent yx 2 4ayP( x1 , y1 ) x 5. Tangents &amp; Normals (ii) Using Cartesian(1) Tangent yx 2 4ayP( x1 , y1 ) x 6. Tangents &amp; Normals (ii) Using Cartesian(1) Tangentx2 yx 2 4ay y 4aP( x1 , y1 ) x 7. Tangents &amp; Normals (ii) Using Cartesian(1) Tangent x2 yx 2 4ayy4ady xdx 2aP( x1 , y1 ) x 8. Tangents &amp; Normals (ii) Using Cartesian(1) Tangent x2 yx 2 4ayy4ady xdx 2a dy x1P( x1 , y1 )when x x1 , dx 2a x 9. Tangents &amp; Normals (ii) Using Cartesian(1) Tangent x2 yx 2 4ayy4ady xdx 2a dy x1P( x1 , y1 )when x x1 , dx 2a xx1 slope of tangent is2a 10. Tangents &amp; Normals (ii) Using Cartesian(1) Tangentx2 yx 2 4ay y 4a dy x dx 2a dy x1P( x1 , y1 )when x x1 , dx 2a xx1 slope of tangent is 2a xy y1 1 x x1 2a 11. Tangents &amp; Normals (ii) Using Cartesian(1) Tangentx2 yx 2 4ay y 4a dy x dx 2a dy x1P( x1 , y1 )when x x1 , dx 2a xx1 slope of tangent is 2a xy y1 1 x x1 2a2ay 2ay1 xx1 x12 12. Tangents &amp; Normals (ii) Using Cartesian(1) Tangentx2 yx 2 4ay y 4a dy x dx 2a dy x1P( x1 , y1 )when x x1 , dx 2a xx1 slope of tangent is 2a xy y1 1 x x1 2a2ay 2ay1 xx1 x122ay 2ay1 xx1 4ay1 13. Tangents &amp; Normals (ii) Using Cartesian(1) Tangentx2 yx 2 4ay y 4a dy x dx 2a dy x1P( x1 , y1 )when x x1 , dx 2a xx1 slope of tangent is 2a xy y1 1 x x1 2a2ay 2ay1 xx1 x122ay 2ay1 xx1 4ay1xx1 2a y y1 14. (2) Normal 15. (2) Normalyx 2 4ayx 16. (2) Normalyx 2 4ay P( x1 , y1 )x 17. (2) Normalyx 2 4ay P( x1 , y1 )x 18. (2) Normalyx 2 4ay P( x1 , y1 )x1 1 Show the slope of tangent at P is 2ax 19. (2) Normalyx 2 4ay P( x1 , y1 )x1 1 Show the slope of tangent at P is 2a2ax2 slope of normal is x1 20. (2) Normalyx 2 4ay P( x1 , y1 )x1 1 Show the slope of tangent at P is 2a2ax2 slope of normal is x1 2a y y1 x x1 x1 21. (2) Normalyx 2 4ay P( x1 , y1 )x1 1 Show the slope of tangent at P is 2a2ax2 slope of normal is x1 2a y y1 x x1 x1x1 y x1 y1 2ax 2ax1 22. (2) Normalyx 2 4ay P( x1 , y1 )x1 1 Show the slope of tangent at P is 2a2ax2 slope of normal is x1 2a y y1 x x1 x1x1 y x1 y1 2ax 2ax12ax x1 y 2ax1 x1 y1 23. (3) Line cutting/touching/missing parabola 24. (3) Line cutting/touching/missing parabolayx 2 4ay x 25. (3) Line cutting/touching/missing parabolayx 2 4ay x 26. (3) Line cutting/touching/missing parabolayx 2 4ay x 27. (3) Line cutting/touching/missing parabolayx 2 4ay y mx b x 28. (3) Line cutting/touching/missing parabola y x 2 4ayparabola and tangent meet when; y mx b x 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. (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 0x 31. (3) Line cutting/touching/missing parabola y x 2 4ay parabola and tangent meet when; y mx bx 2 4a mx b x 2 4amx 4ab 0two solutions (cuts) when 0x 32. (3) Line cutting/touching/missing parabola y x 2 4ay parabola and tangent meet when; y mx bx 2 4a mx b x 2 4amx 4ab 0two solutions (cuts) when 0 one solution (touches) when 0x 33. (3) Line cutting/touching/missing parabola y x 2 4ay parabola and tangent meet when; y mx bx 2 4a mx b x 2 4amx 4ab 0two solutions (cuts) when 0 one solution (touches) when 0x no solutions (misses) when 0 34. (3) Line cutting/touching/missing parabola y x 2 4ay parabola and tangent meet when; y mx bx 2 4a mx b x 2 4amx 4ab 0two solutions (cuts) when 0 one solution (touches) when 0x no solutions (misses) when 0 b2 4ac 35. (3) Line cutting/touching/missing parabola y x 2 4ay parabola and tangent meet when; y mx bx 2 4a mx b x 2 4amx 4ab 0two solutions (cuts) when 0 one solution (touches) when 0x no solutions (misses) when 0 b2 4ac 4am 4 1 4ab 2 36. (3) Line cutting/touching/missing parabola y x 2 4ay parabola and tangent meet when; y mx bx 2 4a mx b x 2 4amx 4ab 0two solutions (cuts) when 0 one solution (touches) when 0x no solutions (misses) when 0 b2 4ac 4am 4 1 4ab 2 16a 2 m 2 16ab 16a am 2 b 37. (3) Line cutting/touching/missing parabola y x 2 4ay parabola and tangent meet when; y mx bx 2 4a mx b x 2 4amx 4ab 0two solutions (cuts) when 0 one solution (touches) when 0x 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. (3) Line cutting/touching/missing parabola y x 2 4ay parabola and tangent meet when; y mx bx 2 4a mx b x 2 4amx 4ab 0two solutions (cuts) when 0 one solution (touches) when 0x 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. (3) Line cutting/touching/missing parabola y x 2 4ay parabola and tangent meet when; y mx bx 2 4a mx b x 2 4amx 4ab 0two solutions (cuts) when 0 one solution (touches) when 0x 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. (3) Line cutting/touching/missing parabola y x 2 4ay parabola and tangent meet when; y mx bx 2 4a mx b x 2 4amx 4ab 0two solutions (cuts) when 0 one solution (touches) when 0x 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. e.g. Find the equation of the two tangents to the parabola x 2 4 y passing through the point (3,2). 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. 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. 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. 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 bb 2 3mtangents are y mx 2 3m 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 bb 2 3mtangents are y mx 2 3m x2 4 y 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 3mtangents are y mx 2 3m x2 4 yx 2 4 mx 2 3m 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 3mtangents are y mx 2 3m x2 4 yx 2 4 mx 2 3m x 2 4mx 12m 8 0 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 3mtangents are y mx 2 3m x2 4 yx 2 4 mx 2 3m x 2 4mx 12m 8 0line is a tangent if 0 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 bb 2 3mtangents are y mx 2 3mx2 4 yx 2 4 mx 2 3m x 2 4mx 12m 8 0 line is a tangent if 0 4m 4 112m 8 02 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 bb 2 3mtangents are y mx 2 3mx2 4 yx 2 4 mx 2 3m x 2 4mx 12m 8 0 line is a tangent if 0 4m 4 112m 8 02 16m2 48m 32 0 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 bb 2 3mtangents are y mx 2 3mx2 4 yx 2 4 mx 2 3m x 2 4mx 12m 8 0 line is a tangent if 0 4m 4 112m 8 02 16m2 48m 32 0m2 3m 2 0 m 1 m 2 0 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 bb 2 3mtangents are y mx 2 3mx2 4 yx 2 4 mx 2 3m x 2 4mx 12m 8 0 line is a tangent if 0 4m 4 112m 8 02 16m2 48m 32 0m2 3m 2 0 m 1 m 2 0m 1 or m 2 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 bb 2 3mtangents are y mx 2 3mx2 4 yx 2 4 mx 2 3m x 2 4mx 12m 8 0 line is a tangent if 0 4m 4 112m 8 02 16m2 48m 32 0m2 3m 2 0 m 1 m 2 0 m 1 or m 2 tangents are y x 1 and y 2 x 4 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 bb 2 3mtangents are y mx 2 3mx2 4 yx 2 4 mx 2 3m x 2 4mx 12m 8 0Exercise 9G; 1ac, 2ac, 3a, 4, 7, 9, 11, 12, line is a tangent if 0 13, 15, 17, 18 4m 4 112m 8 02 16m2 48m 32 0m2 3m 2 0 m 1 m 2 0 m 1 or m 2 tangents are y x 1 and y 2 x 4</p>