1. “Teach A Level Maths” Vol. 2: A2 Core Modules
41: Parametric Equations

2. The Cartesian equation of a curve in a plane is an equation linking x and y.
Some of these equations can be written in a way that is easier to differentiate by using 2 equations, one giving x and one giving y, both in terms of a 3rd variable, the parameter.
Letters commonly used for parameters are s, t and q. ( q is often used if the parameter is an angle. )
e.gs.

3. Converting between Cartesian and Parametric forms
We use parametric equations because they are simpler, so we only convert to Cartesian if asked to do so !
e.g. 1 Change the following to a Cartesian equation and sketch its graph:
Solution: We need to eliminate the parameter t.
We substitute for t from the easier equation:
Subst. in

4. The Cartesian equation is
We usually write this as
Either, we can sketch using a graphical calculator with
and entering the graph in 2 parts.
Or, we can notice that the equation is quadratic with x and y swapped over from the more usual form.

5. Also, the parametric equations
show that as t increases, x increases faster than y.
The sketch is
The curve is called a parabola.

6. e.g. 2 Change the following to a Cartesian equation:
Solution: We need to eliminate the parameter q.
BUT q appears in 2 forms: as and
so, we need a link between these 2 forms.
Which trig identity links and ?
ANS:
To eliminate q we substitute into this expression.

7. So,
becomes
N.B. = not
Multiply by 9:
We have a circle, centre (0, 0), radius 3.

8. Since we recognise the circle in Cartesian form, it’s easy to sketch.
However, if we couldn’t eliminate the parameter or didn’t recognise the curve having done it, we can sketch from the parametric form.

9. e.g. Sketch the curve with equations
Solution:
Let’s see how to do it without eliminating the parameter.
We can easily spot the min and max values of x and y:
( It doesn’t matter that we don’t know which angle q is measuring. )
For both and the min is -1 and the max is +1, so
and
It’s also easy to get the other coordinate at each of these 4 key values e.g.

10. and x
We could draw up a table of values finding x and y for values of q but this is usually very inefficient. Try to just pick out significant features. x x x

11. Think what happens to and as q increases from 0 to . x
We could draw up a table of values finding x and y for values of q but this is usually very inefficient. Try to just pick out significant features. x x x x
Think what happens to and as q increases from 0 to .
This tells us what happens to x and y.

12. Symmetry now completes the diagram.
and x
Symmetry now completes the diagram. x x x x
Think what happens to and as q increases from 0 to .
This tells us what happens to x and y.

13. Symmetry now completes the diagram.
and x
Symmetry now completes the diagram. x x x

14. Symmetry now completes the diagram.
and x
Symmetry now completes the diagram. x x x

15. O The origin is at the centre of the ellipse.x O x
The origin is at the centre of the ellipse. x x x
So, we have the parametric equations of an ellipse
( which we met in Cartesian form in Transformations ).

16. You can use a graphical calculator to sketch curves given in parametric form. However, you will have to use the setup menu before you enter the equations.
You will also have to be careful about the range of values of the parameter and of x and y. If you don’t get the right scales you may not see the whole graph or the graph can be distorted and, for example, a circle can look like an ellipse.
By the time you’ve fiddled around it may have been better to sketch without the calculator!

17. The following equations give curves you need to recognise:
a circle, radius r, centre the origin.
a parabola, passing through the origin, with the x-axis as an axis of symmetry.
an ellipse with centre at the origin, passing through the points (a, 0), (-a, 0), (0, b), (0, -b).

18. To write the ellipse in Cartesian form we use the same trig identity as we used for the circle.
So, for
use
The equation is usually left in this form.

19. There are other parametric equations you might be asked to convert to Cartesian equations. For example, those like the ones in the following exercise.
Exercise
( Use a trig identity ) 1. 2.
Sketch both curves using either parametric or Cartesian equations. ( Use a graphical calculator if you like ).

20. 1.
Solution:
Use
We usually write this in a form similar to the ellipse:
Notice the minus sign. The curve is a hyperbola.

21. Sketch: or
A hyperbola
Asymptotes

22. ( Eliminate t by substitution. )2.
Solution:
Subs. in
The curve is a rectangular hyperbola.

23. Sketch: or
Asymptotes
A rectangular hyperbola.

25. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

26. The Cartesian equation of a curve in a plane is an equation linking x and y.
Some of these equations can be written in a way that is easier to differentiate by using 2 equations, one giving x and one giving y, both in terms of a 3rd variable, the parameter.
Letters commonly used for parameters are s, t and q. ( q is often used if the parameter is an angle. )
e.gs.

27. Converting between Cartesian and Parametric forms
We use parametric equations because they are simpler, so we only convert to Cartesian if asked to do so !
e.g. 1 Change the following to a Cartesian equation and sketch its graph:
Solution: We need to eliminate the parameter t.
Substitution is the easiest way.
Subst. in

28. The Cartesian equation is
We usually write this as
Either, we can sketch using a graphical calculator with
and entering the graph in 2 parts.
Or, we can notice that the equation is quadratic with x and y swapped over from the more usual form.

29. e.g. 2 Change the following to a Cartesian equation:
Solution: We need to eliminate the parameter q.
BUT q appears in 2 forms: as and
so, we need a link between these 2 forms.
To eliminate q we substitute into the expression.

30. Multiply by 9:
becomes
So,
N.B. = not
We have a circle, centre (0, 0), radius 3.

31. The following equations give curves you need to recognise:
a circle, radius r, centre the origin.
a parabola, passing through the origin, with the x-axis an axis of symmetry.
an ellipse with centre at the origin, passing through the points (a, 0), (-a, 0), (0, b), (0, -b).