Parametric Equations - Eliminating the Parameter (CIE A Level Maths: Pure 3)

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Parametric Equations - Eliminating the Parameter

What does eliminating the parameter mean? 

Notes para_cart, AS & A Level Maths revision notes

  • In parametric equations, x = f(t) and y = g(t)
  • There is still a connection directly linking x and y
    • This will be the Cartesian equation of the graph

How do I find the Cartesian equation from parametric equations? 

Notes elim_t, AS & A Level Maths revision notes

  • STEP 1: Rearrange one of the equations to make t the subject
    • Either t = p(x) or t = q(y)
  • STEP 2: Substitute into the other equation
  • STEP 3 Rearrange into the desired (Cartesian) form

How do I eliminate t when trig is involved?

Notes elim_trig_t, AS & A Level Maths revision notes

  • STEP 1  Rearrange both equations into the forms “cos t = …” and “sin t = …”
  • STEP 2  Square BOTH sides of BOTH equations
  • STEP 3  Add the equations together
  • STEP 4  The trig identity “sin2 x + cos2 x 1” eliminates t
  • STEP 5  Rearrange into desired (Cartesian) form
    • This technique is seen in Trigonometric Identities

Examiner Tip

When choosing which equation to rearrange, aim for “as simple as possible”:

  • Linear equations are simpler than quadratics
    • eg Rearrange x = 2t + 3

      or

      y = 3t2 +3t -4 ?

  • Single exponential terms are quite easy to deal with
    • eg x = et t = ln x

Trig identities may be needed and remember squared terms are good!

  • eg sin2 x + cos2 x 1

Worked example

Example soltn, AS & A Level Maths revision notes

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.