Eliminating the Parameter of Parametric Equations (Cambridge International AS Mathematics)

Parametric Equations - Eliminating the Parameter

Eliminating the parameter of parametric equations

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 can I eliminate the parameter using trigonometric identities?

• 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 “sin²x + cos²x ≡ 1” eliminates t

• STEP 5 Rearrange into desired (Cartesian) form

  • This technique is seen in Trigonometric Identities

Exam Hint 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 = e^t→ t = ln x

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

• eg sin²x + cos²x ≡ 1

Worked Example