A Level Maths: Parametric Equations - Finding The Cartesian Equation
Introduction
You have seen an equation such as 3x + y = 7. The fact that it is in terms of x and y is referred to as a cartesian equation.
A parametric equation is defined in terms of third variable often t and sometimes θ.
So:
would be an example of a parametric equation.
Quite often when given a set of parametric equations, the cartesian equation needs to be found.
Example
Given the parametric equations:
Determine the cartesian equation.
Method: In order to answer this question the key is to make t the subject from the simpler equation and to substitute this into the more complicated equation.
Making t the subject from the simpler equation gives:
Now substituting this into the second equation will give:
And this will do as an answer. There is no need to simplify the expression unless a question requires you to do so.
This was a more simple example. Questions can be more algebraic intense as the following example will show.
Example
Given the parametric equations:
Determine the cartesian equation.
Method: In the last example t was made the subject and substituted into the more complicated equation. However, in this case, neither equation is more simpler than the other, so we will start with the first equation to make t the subject.
This is simply a case of rearranging formulas, a topic covered at GCSE level.
Now we need to substitute this into the second equation which will give:
This needs to be made simpler!
What we have here is a fraction over a fraction. The more complicated fraction is the bottom one so we will first write this as a single algebraic fraction.
That is the second fraction written as a single algebraic fraction. Now we can go back to y = ….
Remember a fraction is simply a division. What you should notice is that the denominator of each fraction is 1 – x. Because they are both the same, they can now cancel each other out.
This is the cartesian equation.
Alternative Method: As an alternative to the above method you can also make t the subject.
From the first equation we already have:
From the second equation making t the subject will give you:
Now because both are equal to t we can equate them and make y the subject as follows:
And this is the same answer as before.
θθ
Finding finding a cartesian equation involving trigonometric terms
Example 3
Suppose we want to find the cartesian of the following parametric equations:
From what we have seen already, it would suggest that we would make the subject. From the first equation doing this would give:
Which just by looking at, is not going to be easy to work with. So this is not a viable option.
When it comes to parametric equations involving trigonometric terms it is important to be able to determine a relationship that exists between them and this can be done using the trigonometric identities.
For the second equation we have a double angle trig term.
As a part of the A Level Maths content in year 13 you ought to have met what is meant by a double angle. For the above we have:
Now let us try and establish that relationship between the parametric equations that we have:
The challenge with these questions is being able to identify which is the correct trig identity to use. This will come with practice and needs to be a structured part of your A Level Maths Revision.
Example 4:
Find the cartesian cartesian equation defined by the following parametric equations:
b) Hence sketch the curve
Method: Once again we have a set of parametric equations that are defined in terms of trigonometry and what is needed is to determine the relationship that exists between cosθ and sinθ.
From Year 12 you should have been introduced to the following trig identity:
And this is the cartesian equation.
b) In order to sketch the curve we will have the following points where it cross the axes as follows:
x= 4 and y = 3
Don’t forget that on the x-axis y = 0 and on the y-axis . Having this information allows us to sketch the curve.
And what we have here is an ellipse.
And this is the cartesian equation.
Example 5
Given the parametric equations:
Find the cartesian equation.
State the type of curve and any properties that it may have.
Method: Once again we need to find a relationship that exists between cosθ and sinθ.
The most suitable trig identity that gives a relationship between these two trig terms is:
b) You can now recognise that this forms the equation of a circle with centre 2, -1 and radius = 3.
Summary
Being able to find the cartesian equation from a set of parametric equations is a valuable skill and one that requires a good level of practice.
In a future article we will look at Parametric Differentiation which relies on using the Chain Rule and there are some exam boards that ask questions regarding Parametric Integration.
If you are having some problems with parametric equations then you should consider attending our upcoming Christmas A Level Maths Revision Course which takes place in Central London between the 27th – 29th December 2023 and 2nd – 4th January 2024.
The course is for 3 days duration with a one hour assessment with written feedback to highlight any areas which may be costing your valuable marks so allowing you to perform to your best ability in any upcoming January mock exams.
Whatever your goals if you need help getting those top grades then just complete the form and we will be in contact within 24 hours.