Pearson Edexcel · A-Level · Mathematics
Course version: 9MA0 · Version from Mar 2026
Understand and use the parametric equations of curves and conversion…
Full specification wording
Coordinate geometry in the (x, y) plane: 3.3 Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms.
Parametric equations define x and y separately in terms of a third variable, the parameter, such as x = 2t and y = t² − 1.
Checked against the Pearson Edexcel 9MA0 specification (opens in a new tab) on .
- Point 3 of 4
- 2 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
Parametric equations define x and y separately in terms of a third variable, the parameter, such as x = 2t and y = t² − 1. Each value of t gives one point, and the set of points traces the curve. To convert to Cartesian form, eliminate t: here t = x/2, so y = (x/2)² − 1 = x²/4 − 1, a parabola. To convert the other way, choose a parameter, often t = x or a trigonometric identity. For example, a circle x² + y² = r² becomes x = r cos θ, y = r sin θ because cos²θ + sin²θ = 1. You may also need to restrict the parameter or the domain so the parametric form matches the intended part of the curve.
What earns the marks
- Substitutes correctly from one parametric equation into the other to eliminate the parameter.
- Uses a trigonometric identity such as cos²θ + sin²θ = 1 to convert between Cartesian and parametric forms.
- Chooses a suitable parameter when converting a Cartesian equation to parametric form and states it clearly.
- States any restriction on the parameter or on x and y so the parametric form represents the correct portion of the curve.
- Finds coordinates of points on the curve at a given parameter value by substituting into the parametric equations.
Common mistakes and how to put them right
Eliminating the parameter incorrectly by adding equations that should be substituted.
Put it right: Express t from one equation and substitute into the other.
Losing the domain restriction.
Put it right: State the range of the parameter.
Forgetting the identity when converting a circle.
Put it right: Use x = r cos θ and y = r sin θ so that cos²θ + sin²θ = 1 holds.
Exam tips
- Write the parameter clearly and show each substitution step.
- After converting, test one parameter value in both forms to check they agree.
Try the lesson
The start of the Coordinate geometry in the (x, y) plane: 3.3 Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms. lesson, made from checked teaching notes for this specification point.
Have a go first
A curve has parametric equations x = 3t and y = t². Find the Cartesian equation of the curve by eliminating t.
Listen
Parametric equations define x and y separately in terms of a third variable, the parameter, such as x = 2t and y = t² − 1. Each value of t gives one point, and the set of points traces the curve. To convert to Cartesian form, eliminate t: here t = x/2, so y = (x/2)² − 1 = x²/4 − 1, a parabola. To convert the other way, choose a parameter, often t = x or a trigonometric identity. For example, a circle x² + y² = r² becomes x = r cos θ, y = r sin θ because cos²θ + sin²θ = 1. You may also need to restrict the parameter or the domain so the parametric form matches the intended part of the curve. A strong answer earns marks for these points. Substitutes correctly from one parametric equation into the other to eliminate the parameter. Uses a trigonometric identity such as cos²θ + sin²θ = 1 to convert between Cartesian and parametric forms. Chooses a suitable parameter when converting a Cartesian equation to parametric form and states it clearly. States any restriction on the parameter or on x and y so the parametric form represents the correct portion of the curve. A common slip: Eliminating the parameter incorrectly by adding equations that should be substituted. Instead, express t from one equation and substitute into the other. Exam tip: Write the parameter clearly and show each substitution step.
Key words
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Parameterisation
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Eliminating the parameter
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See every definition
- Parameterisation
- Defining x and y separately in terms of a third variable, the parameter, such as x = 2t and y = t² − 1.
- Eliminating the parameter
- Converting to Cartesian form by expressing the parameter from one equation and substituting into the other.
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