Coordinate geometry in the (x, y) plane (A2 Unit 3: Pure Mathematics B) — WJEC A-Level Mathematics
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Coordinate geometry in the (x, y) plane (A2 Unit 3: Pure Mathematics B) explained
Parametric equations express the Cartesian coordinates x and y independently as functions of a third variable or parameter t, so x = f(t) and y = g(t).
Read the full explanation
Conversion to Cartesian form y = F(x) eliminates the parameter t. When equations are algebraic, rearrange one equation to isolate t (e.g. t = (x − a)/b) and substitute into the other equation. When equations involve trigonometric functions of t, eliminate t using fundamental identities; for example, x = a cos t and y = b sin t rearrange to (x/a)² + (y/b)² = cos²t + sin²t = 1, defining an ellipse. In calculus, the chain rule gives the parametric derivative dy/dx = (dy/dt) / (dx/dt), which enables finding tangents, normals, and stationary points without converting to Cartesian form.
Your focus
- Convert parametric equations into Cartesian form by algebraic substitution and trigonometric identities.
- Determine the Cartesian domain and range from given parametric intervals.
- Differentiate parametric equations using dy/dx = (dy/dt) / (dx/dt).
Coordinate geometry in the (x, y) plane (A2 Unit 3: Pure Mathematics B) exam tips
Marking Points
- eliminating the parameter t algebraically or using trigonometric identities
- stating the resulting Cartesian equation in specified form
- finding the Cartesian domain and range from the given parameter interval
- calculating the gradient dy/dx using the parametric chain rule (dy/dt) / (dx/dt)
Examiner Tips
- 💡Use trigonometric identities like cos²t + sin²t = 1 or sec²t − tan²t = 1 to eliminate trigonometric parameters.
- 💡Find the domain of the Cartesian equation by calculating the minimum and maximum values of x(t).
Common Mistakes
- making algebraic errors when eliminating t, especially when squaring trigonometric functions
- forgetting that the domain of the Cartesian curve corresponds to the range of x(t) over the parameter interval