Topic 17 Parametric representation

Sometimes the variables \(x\) and \(y\) are themselves functions of another variable, called parameter, say \(x=x(t)\) and \(y=y(t)\). We may be able to express \(y\) in terms of \(x\) by eliminating \(t\).

Example 17.1 \(x=a\cos\theta\),   \(y=a\sin\theta\), where \(a\) is a positive constant and \(\theta\) is the parameter.

We aim to eliminate \(\theta\) and express \(y\) in terms of \(x\).
We square both equations and add them: \(x^2=a^2\sin^2\theta\),   \(y^2=a^2\cos^2\theta\)
\(\implies\) \(x^2+y^2=a^2(\cos^2\theta+\sin^2\theta) = a^2\)
\(\implies\) \(y=\pm\sqrt{a^2-x^2}\)   (thus in this case, there are two “branches” of the function: one with the “+” and one with the “-”; see Figure 17.1).

Graph of ±√(4-x²)

Figure 17.1: Graph of ±√(4-x²)

Example 17.2 A curve is given by \(x=t^2\) and \(y=2t\). Find its Cartesian equation.

Solution. Calculate

  1. \(y=2t\) \(\implies\) \(t^2=\frac{y^2}{4}\)
  2. \(x=t^2\)

Putting 1. and 2. together, we get \(x=\frac{y^2}{4}\) \(\iff\) \(y=\pm2\sqrt{x}\). See Figure 17.2.

Graph of ±2√x

Figure 17.2: Graph of ±2√x