1. Tangents & Normals
(ii) Using Cartesian

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

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

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

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

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

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

8. Tangents & Normals
(ii) Using Cartesian
y
x
2
4x ay
1 1( , )P x y
(1) Tangent
a
x
y
4
2

a
x
dx
dy
2

1
1when ,
2
xdy
x x
dx a
 

9. Tangents & Normals
(ii) Using Cartesian
y
x
2
4x ay
1 1( , )P x y
(1) Tangent
a
x
y
4
2

a
x
dx
dy
2

1
1when ,
2
xdy
x x
dx a
 
1
slope of tangent is
2
x
a


10. Tangents & Normals
(ii) Using Cartesian
y
x
2
4x ay
1 1( , )P x y
(1) Tangent
a
x
y
4
2

a
x
dx
dy
2

1
1when ,
2
xdy
x x
dx a
 
1
slope of tangent is
2
x
a

 1
1
1
2
xx
a
x
yy 

11. Tangents & Normals
(ii) Using Cartesian
y
x
2
4x ay
1 1( , )P x y
(1) Tangent
a
x
y
4
2

a
x
dx
dy
2

1
1when ,
2
xdy
x x
dx a
 
1
slope of tangent is
2
x
a

 1
1
1
2
xx
a
x
yy 
2
11122 xxxayay 

12. Tangents & Normals
(ii) Using Cartesian
y
x
2
4x ay
1 1( , )P x y
(1) Tangent
a
x
y
4
2

a
x
dx
dy
2

1
1when ,
2
xdy
x x
dx a
 
1
slope of tangent is
2
x
a

 1
1
1
2
xx
a
x
yy 
2
11122 xxxayay 
111 422 ayxxayay 

13. Tangents & Normals
(ii) Using Cartesian
y
x
2
4x ay
1 1( , )P x y
(1) Tangent
a
x
y
4
2

a
x
dx
dy
2

1
1when ,
2
xdy
x x
dx a
 
1
slope of tangent is
2
x
a

 1
1
1
2
xx
a
x
yy 
2
11122 xxxayay 
111 422 ayxxayay 
 11 2 yyaxx 

14. (2) Normal

15. (2) Normal
y
x
2
4x ay

16. (2) Normal
y
x
2
4x ay
1 1( , )P x y

17. (2) Normal
y
x
2
4x ay
1 1( , )P x y

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

19. (2) Normal
y
x
2
4x ay
1 1( , )P x y 1 1
Show the slope of tangent at is
2
x
P
a
2
1
2
slope of normal is
a
x
 

20. (2) Normal
y
x
2
4x ay
1 1( , )P x y 1 1
Show the slope of tangent at is
2
x
P
a
2
1
2
slope of normal is
a
x
 
 1
1
1
2
xx
x
a
yy 



21. (2) Normal
y
x
2
4x ay
1 1( , )P x y 1 1
Show the slope of tangent at is
2
x
P
a
2
1
2
slope of normal is
a
x
 
 1
1
1
2
xx
x
a
yy 


1111 22 axaxyxyx 

22. (2) Normal
y
x
2
4x ay
1 1( , )P x y 1 1
Show the slope of tangent at is
2
x
P
a
2
1
2
slope of normal is
a
x
 
 1
1
1
2
xx
x
a
yy 


1111 22 axaxyxyx 
1111 22 yxaxyxax 

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

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

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

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

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

28. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b 
parabola and tangent meet when;

29. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b   2
4x a mx b 
parabola and tangent meet when;

30. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b   2
4x a mx b 
parabola and tangent meet when;
2
4 4 0x amx ab  

31. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b   2
4x a mx b 
parabola and tangent meet when;
2
4 4 0x amx ab  
two solutions (cuts) when 0 

32. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b   2
4x a mx b 
parabola and tangent meet when;
2
4 4 0x amx ab  
two solutions (cuts) when 0 
one solution (touches) when 0 

33. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b   2
4x a mx b 
parabola and tangent meet when;
2
4 4 0x amx ab  
two solutions (cuts) when 0 
one solution (touches) when 0 
no solutions (misses) when 0 

34. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b   2
4x a mx b 
2
4b ac  
parabola and tangent meet when;
2
4 4 0x amx ab  
two solutions (cuts) when 0 
one solution (touches) when 0 
no solutions (misses) when 0 

35. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b   2
4x a mx b 
2
4b ac  
    
2
4 4 1 4am ab   
parabola and tangent meet when;
2
4 4 0x amx ab  
two solutions (cuts) when 0 
one solution (touches) when 0 
no solutions (misses) when 0 

36. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b   2
4x a mx b 
2
4b ac  
    
2
4 4 1 4am ab   
 
2 2
2
16 16
16
a m ab
a am b
 
 
parabola and tangent meet when;
2
4 4 0x amx ab  
two solutions (cuts) when 0 
one solution (touches) when 0 
no solutions (misses) when 0 

37. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b   2
4x a mx b 
2
4b ac  
    
2
4 4 1 4am ab   
 
2 2
2
16 16
16
a m ab
a am b
 
 
parabola and tangent meet when;
2
4 4 0x amx ab  
two solutions (cuts) when 0 
one solution (touches) when 0 
no solutions (misses) when 0 
2
two solutions (cuts) when 0am b  

38. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b   2
4x a mx b 
2
4b ac  
    
2
4 4 1 4am ab   
 
2 2
2
16 16
16
a m ab
a am b
 
 
parabola and tangent meet when;
2
4 4 0x amx ab  
two solutions (cuts) when 0 
one solution (touches) when 0 
no solutions (misses) when 0 
2
two solutions (cuts) when 0am b  
2
one solution (touches) when 0am b 

39. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b   2
4x a mx b 
2
4b ac  
    
2
4 4 1 4am ab   
 
2 2
2
16 16
16
a m ab
a am b
 
 
parabola and tangent meet when;
2
4 4 0x amx ab  
two solutions (cuts) when 0 
one solution (touches) when 0 
no solutions (misses) when 0 
2
two solutions (cuts) when 0am b  
2
one solution (touches) when 0am b  (common idea)

40. (3) Line cutting/touching/missing parabola
y
x
2
4x ay
y mx b   2
4x a mx b 
2
4b ac  
    
2
4 4 1 4am ab   
 
2 2
2
16 16
16
a m ab
a am b
 
 
parabola and tangent meet when;
2
4 4 0x amx ab  
two solutions (cuts) when 0 
one solution (touches) when 0 
no solutions (misses) when 0 
2
two solutions (cuts) when 0am b  
2
one solution (touches) when 0am b 
2
no solutions (misses) when 0am b 
(common idea)

41. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y

42. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b

43. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
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
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b
2 3m b  
2 3b m 

45. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b
2 3m b  
2 3b m 
tangents are 2 3y mx m  

46. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b
2 3m b  
2 3b m 
tangents are 2 3y mx m  
2
4x y

47. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b
2 3m b  
2 3b m 
tangents are 2 3y mx m  
2
4x y
 2
4 2 3x mx m  

48. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b
2 3m b  
2 3b m 
tangents are 2 3y mx m  
2
4x y
 2
4 2 3x mx m  
 2
4 12 8 0x mx m   

49. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b
2 3m b  
2 3b m 
tangents are 2 3y mx m  
2
4x y
 2
4 2 3x mx m  
 2
4 12 8 0x mx m   
line is a tangent if 0 

50. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b
2 3m b  
2 3b m 
tangents are 2 3y mx m  
2
4x y
 2
4 2 3x mx m  
 2
4 12 8 0x mx m   
line is a tangent if 0 
    
2
4 4 1 12 8 0m m   

51. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b
2 3m b  
2 3b m 
tangents are 2 3y mx m  
2
4x y
 2
4 2 3x mx m  
 2
4 12 8 0x mx m   
line is a tangent if 0 
    
2
4 4 1 12 8 0m m   
2
16 48 32 0m m  

52. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b
2 3m b  
2 3b m 
tangents are 2 3y mx m  
2
4x y
 2
4 2 3x mx m  
 2
4 12 8 0x mx m   
line is a tangent if 0 
    
2
4 4 1 12 8 0m m   
2
16 48 32 0m m  
  
2
3 2 0
1 2 0
m m
m m
  
  

53. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b
2 3m b  
2 3b m 
tangents are 2 3y mx m  
2
4x y
 2
4 2 3x mx m  
 2
4 12 8 0x mx m   
line is a tangent if 0 
    
2
4 4 1 12 8 0m m   
2
16 48 32 0m m  
  
2
3 2 0
1 2 0
m m
m m
  
  
1 or 2m m 

54. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b
2 3m b  
2 3b m 
tangents are 2 3y mx m  
2
4x y
 2
4 2 3x mx m  
 2
4 12 8 0x mx m   
line is a tangent if 0 
    
2
4 4 1 12 8 0m m   
2
16 48 32 0m m  
  
2
3 2 0
1 2 0
m m
m m
  
  
1 or 2m m 
tangents are 1 and 2 4y x y x    

55. e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2
4x y
tangent will be of the form y = mx + b
2 3m b  
2 3b m 
tangents are 2 3y mx m  
2
4x y
 2
4 2 3x mx m  
 2
4 12 8 0x mx m   
line is a tangent if 0 
    
2
4 4 1 12 8 0m m   
2
16 48 32 0m m  
  
2
3 2 0
1 2 0
m m
m m
  
  
1 or 2m m 
tangents are 1 and 2 4y x y x    
Exercise 9G; 1ac, 2ac,
3a, 4, 7, 9, 11, 12,
13, 15, 17, 18