Exercises 17 (Parametric representation)

  1. Plot the functions given below in parametric form for the range \(0\) to \(2\pi\,\mathrm{rad}\):

    1. \(x = 10\cos\theta\), \(y = 10\sin\theta\)
    2. \(x = 10\cos\theta\), \(y = 5\sin\theta\)

  2. Let \(t\) vary from \(-4\) to \(+4\) for \(x = 1 - t\), \(y = t\) and plot the function. Also eliminate the variable \(t\) to express \(x\) in terms of \(y\).

  3. For the equations below, stated in parametric form, eliminate the parameter \(t\) to obtain the Cartesian form. Assume \(a\) and \(c\) are constants.

    1. \(x = at^2\),     \(y = 2at\)
    2. \(x = ct\),     \(y=\frac{c}{t}\)
    3. \(x = 2t + 1\),     \(y = 2t(t - 1)\)

Solutions: 2. \(y=1-x\); 3. (i) \(y^2=4ax\); (ii) \(y=\frac{c^2}{x}\); (iii) \(y=\frac{x^2-4x+3}{2}\)