CIE A Level Maths: Pure 3复习笔记4.3.2 Parametric Equations - Eliminating the Parameter

Parametric Equations - Eliminating the Parameter

What does eliminating the parameter mean? 


  • 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? 


  • 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?


  • 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

Exam Tip

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

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

      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