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Parametric Equations - Eliminating the Parameter (CIE A Level Maths: Pure 3)
Revision Note
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
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 ?
- eg Rearrange x = 2t + 3
- 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
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