C: Coordinate geometry in the (x, y) plane: Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms.

The idea

Parametric equations describe a curve using a third variable, usually t or θ. For example, x = t², y = 2t defines a parabola. To convert to Cartesian form, eliminate the parameter. From y = 2t, t = y/2, so x = (y/2)² = y²/4, giving y² = 4x. Conversely, to parametrise a curve, choose a suitable parameter. For a circle x² + y² = r², use x = r cos θ, y = r sin θ. You need to understand how parameters work, sketch curves by plotting points for varying parameter values, and convert between forms. This skill is essential for linking algebraic and geometric representations and understanding the domain and range implications of the chosen parameter.

What earns the marks

• Eliminate the parameter to convert parametric equations into a Cartesian equation.
• Choose appropriate parametric equations for a given Cartesian curve, such as using trigonometric functions for circles and ellipses.
• Determine points of intersection between a parametric curve and a line or another curve by substituting parametric forms into the Cartesian equation.
• Sketch a curve from its parametric equations by considering key points, domain restrictions, and behaviour as the parameter varies.

Common mistakes and how to put them right

• Eliminating the parameter incorrectly, such as squaring only one side of an equation or losing solutions.

  Put it right: Carefully manipulate the equations to express t in terms of x or y, then substitute; check for domain restrictions.

• Forgetting that parametric equations can have restrictions on the parameter that affect the Cartesian equation.

  Put it right: Consider the range of the parameter and how it maps to the Cartesian curve.

• Failing to use trigonometric identities when converting trigonometric parametric equations.

  Put it right: Use identities like cos²θ + sin²θ = 1 to eliminate θ instead of trying to use inverse trigonometric functions.

Exam tips

• When converting from parametric to Cartesian, look for a trigonometric identity or a simple algebraic relationship to eliminate the parameter.
• If asked to sketch a parametric curve, plot points for several values of the parameter and join them smoothly, noting any symmetries.
• Check whether the parameter has a restricted domain; this may affect the Cartesian equation or the sketch.