The document discusses parametric coordinates and compares them to Cartesian coordinates. It states that parametric coordinates describe a curve using two equations where points are defined by a single parameter value, while Cartesian coordinates use a single equation where points are defined by two number values. The document then provides an example of a parabola with the parametric coordinates x=2at, y=at^2 and discusses how to derive the Cartesian equation from the parametric form. It also identifies the focus of the parabola as (1/4,0) and calculates the parametric coordinates for the curve y=8x^2 as (t/16, t^2/32).

1. Parametric Coordinates

2. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.

3. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.
Parametric Coordinates: curve is described by two equations and points
are described by one number (parameter).

4. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.
Parametric Coordinates: curve is described by two equations and points
are described by one number (parameter).
y
x

5. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.
Parametric Coordinates: curve is described by two equations and points
are described by one number (parameter).
y
x

6. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.
Parametric Coordinates: curve is described by two equations and points
are described by one number (parameter).
y x 2  4ay
x

7. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.
Parametric Coordinates: curve is described by two equations and points
are described by one number (parameter).
y x 2  4ay Cartesian equation
x

8. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.
Parametric Coordinates: curve is described by two equations and points
are described by one number (parameter).
y x 2  4ay Cartesian equation
x  2at , y  at 2
x

9. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.
Parametric Coordinates: curve is described by two equations and points
are described by one number (parameter).
y x 2  4ay Cartesian equation
x  2at , y  at 2 Parametric coordinates
x

10. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.
Parametric Coordinates: curve is described by two equations and points
are described by one number (parameter).
y x 2  4ay Cartesian equation
x  2at , y  at 2 Parametric coordinates
(2a, a)
x

11. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.
Parametric Coordinates: curve is described by two equations and points
are described by one number (parameter).
y x 2  4ay Cartesian equation
x  2at , y  at 2 Parametric coordinates
(2a, a) Cartesian coordinates
x

12. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.
Parametric Coordinates: curve is described by two equations and points
are described by one number (parameter).
y x 2  4ay Cartesian equation
x  2at , y  at 2 Parametric coordinates
(2a, a) Cartesian coordinates
t 1
x

13. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.
Parametric Coordinates: curve is described by two equations and points
are described by one number (parameter).
y x 2  4ay Cartesian equation
x  2at , y  at 2 Parametric coordinates
(2a, a) Cartesian coordinates
t 1 parameter
x

14. Parametric Coordinates
Cartesian Coordinates: curve is described by one equation and points
are described by two numbers.
Parametric Coordinates: curve is described by two equations and points
are described by one number (parameter).
y x 2  4ay Cartesian equation
(4a, 4a) x  2at , y  at 2 Parametric coordinates
t  2
(2a, a) Cartesian coordinates
t 1 parameter
x

15. Any point on the parabola x 2  4ay has coordinates;

16. Any point on the parabola x 2  4ay has coordinates;
x  2at

17. Any point on the parabola x 2  4ay has coordinates;
x  2at y  at 2

18. Any point on the parabola x 2  4ay has coordinates;
x  2at y  at 2
where; a is the focal length

19. Any point on the parabola x 2  4ay has coordinates;
x  2at y  at 2
where; a is the focal length
t is any real number

20. Any point on the parabola x 2  4ay has coordinates;
x  2at y  at 2
where; a is the focal length
t is any real number
e.g. Eliminate the parameter to find the cartesian equation of;
1 1
x  t , y  t2
2 4

21. Any point on the parabola x 2  4ay has coordinates;
x  2at y  at 2
where; a is the focal length
t is any real number
e.g. Eliminate the parameter to find the cartesian equation of;
1 1
x  t , y  t2
2 4
t  2x

22. Any point on the parabola x 2  4ay has coordinates;
x  2at y  at 2
where; a is the focal length
t is any real number
e.g. Eliminate the parameter to find the cartesian equation of;
1 1
x  t , y  t2
2 4
1
y   2x
2
t  2x
4

23. Any point on the parabola x 2  4ay has coordinates;
x  2at y  at 2
where; a is the focal length
t is any real number
e.g. Eliminate the parameter to find the cartesian equation of;
1 1
x  t , y  t2
2 4
1
y   2x
2
t  2x
4
y   4x2 
1
4
y  x2

24. Any point on the parabola x 2  4ay has coordinates;
x  2at y  at 2
where; a is the focal length
t is any real number
e.g. Eliminate the parameter to find the cartesian equation of;
1 1
x  t , y  t2
2 4
1
y   2x
2
t  2x
4
y   4x2 
1
4
y  x2
(ii) State the coordinates of the focus

25. Any point on the parabola x 2  4ay has coordinates;
x  2at y  at 2
where; a is the focal length
t is any real number
e.g. Eliminate the parameter to find the cartesian equation of;
1 1
x  t , y  t2
2 4
1
y   2x
2
t  2x
4
y   4x2 
1
4
y  x2
(ii) State the coordinates of the focus
1
a
4

26. Any point on the parabola x 2  4ay has coordinates;
x  2at y  at 2
where; a is the focal length
t is any real number
e.g. Eliminate the parameter to find the cartesian equation of;
1 1
x  t , y  t2
2 4
1
y   2x
2
t  2x
4
y   4x2 
1
4
y  x2
(ii) State the coordinates of the focus  1
a
1  focus   0, 
4  4

27. (iii) Calculate the parametric coordinates of the curve y  8 x 2

28. (iii) Calculate the parametric coordinates of the curve y  8 x 2
x 2  4ay

29. (iii) Calculate the parametric coordinates of the curve y  8 x 2
x 2  4ay
1
4a 
8
1
a
32

30. (iii) Calculate the parametric coordinates of the curve y  8 x 2
x 2  4ay
1
4a 
8
1
a
32
1 1 
 the parametric coordinates are  t , t 2 
 16 32 

31. (iii) Calculate the parametric coordinates of the curve y  8 x 2
x 2  4ay
1
4a 
8
1
a
32
1 1 
 the parametric coordinates are  t , t 2 
 16 32 
Exercise 9D; 1, 2 (not latus rectum), 3, 5, 7a