1. Tangents & Normals
(ii) Using Cartesian

2. Tangents & Normals
(ii) Using Cartesian
(1) Tangent

3. Tangents & Normals
(ii) Using Cartesian
(1) Tangent
y x 2  4ay
x

4. Tangents & Normals
(ii) Using Cartesian
(1) Tangent
y x 2  4ay
P( x1 , y1 )
x

5. Tangents & Normals
(ii) Using Cartesian
(1) Tangent
y x 2  4ay
P( x1 , y1 )
x

6. Tangents & Normals
(ii) Using Cartesian
(1) Tangent x2
y x 2  4ay y
4a
P( x1 , y1 )
x

7. Tangents & Normals
(ii) Using Cartesian
(1) Tangent x2
y x 2  4ay y
4a
dy x

dx 2a
P( x1 , y1 )
x

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. 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. 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. 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. 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. 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. (2) Normal

15. (2) Normal
y x 2  4ay
x

16. (2) Normal
y x 2  4ay
P( x1 , y1 )
x

17. (2) Normal
y x 2  4ay
P( x1 , y1 )
x

18. (2) Normal
y x 2  4ay
P( x1 , y1 ) x1
1 Show the slope of tangent at P is
2a
x

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. (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. (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. (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. (3) Line cutting/touching/missing parabola

24. (3) Line cutting/touching/missing parabola
y x 2  4ay
x

25. (3) Line cutting/touching/missing parabola
y x 2  4ay
x

26. (3) Line cutting/touching/missing parabola
y x 2  4ay
x

27. (3) Line cutting/touching/missing parabola
y x 2  4ay
y  mx  b
x

28. (3) Line cutting/touching/missing parabola
y x 2  4ay parabola 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  0
x

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. (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. (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. (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. (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. (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. (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. (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. (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. (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. 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  b
b  2  3m
tangents 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  b
b  2  3m
tangents 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  3m
tangents are y  mx  2  3m
x2  4 y
x 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  3m
tangents are y  mx  2  3m
x2  4 y
x 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  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. 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. 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. 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. 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. 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. 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