Convert to Cartesian (polynomial/rational)

Questions asking to eliminate the parameter from polynomial or rational parametric equations to obtain a Cartesian equation y = f(x) or implicit form, where trigonometric identities are not the primary method.

33 questions · Moderate -0.2

Sort by: Default | Easiest first | Hardest first
AQA C4 2006 January Q2
11 marks Moderate -0.3
2 A curve is defined by the parametric equations $$x = 3 - 4 t \quad y = 1 + \frac { 2 } { t }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the equation of the tangent to the curve at the point where \(t = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Verify that the cartesian equation of the curve can be written as $$( x - 3 ) ( y - 1 ) + 8 = 0$$
AQA C4 2007 January Q1
11 marks Moderate -0.8
1 A curve is defined by the parametric equations $$x = 1 + 2 t , \quad y = 1 - 4 t ^ { 2 }$$
    1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t }\).
      (2 marks)
    2. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find an equation of the normal to the curve at the point where \(t = 1\).
  3. Find a cartesian equation of the curve.
AQA C4 2008 January Q5
10 marks Standard +0.3
5 A curve is defined by the parametric equations \(x = 2 t + \frac { 1 } { t ^ { 2 } } , \quad y = 2 t - \frac { 1 } { t ^ { 2 } }\).
  1. At the point \(P\) on the curve, \(t = \frac { 1 } { 2 }\).
    1. Find the coordinates of \(P\).
    2. Find an equation of the tangent to the curve at \(P\).
  2. Show that the cartesian equation of the curve can be written as $$( x - y ) ( x + y ) ^ { 2 } = k$$ where \(k\) is an integer.
AQA C4 2005 June Q5
10 marks Moderate -0.8
5 A curve is defined by the parametric equations $$x = 2 t + \frac { 1 } { t } , \quad y = \frac { 1 } { t } , \quad t \neq 0$$
  1. Find the coordinates of the point on the curve where \(t = \frac { 1 } { 2 }\).
  2. Show that the cartesian equation of the curve can be written as $$x y - y ^ { 2 } = 2$$
  3. Show that the gradient of the curve at the point \(( 3,2 )\) is 2 .
AQA C4 2008 June Q2
10 marks Moderate -0.3
2 A curve is defined, for \(t \neq 0\), by the parametric equations $$x = 4 t + 3 , \quad y = \frac { 1 } { 2 t } - 1$$ At the point \(P\) on the curve, \(t = \frac { 1 } { 2 }\).
  1. Find the gradient of the curve at the point \(P\).
  2. Find an equation of the normal to the curve at the point \(P\).
  3. Find a cartesian equation of the curve.
AQA C4 2009 June Q2
11 marks Standard +0.3
2 A curve is defined by the parametric equations $$x = \frac { 1 } { t } , \quad y = t + \frac { 1 } { 2 t }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find an equation of the normal to the curve at the point where \(t = 1\).
  3. Show that the cartesian equation of the curve can be written in the form $$x ^ { 2 } - 2 x y + k = 0$$ where \(k\) is an integer.
AQA Paper 2 2020 June Q8
10 marks Standard +0.3
8 The curve defined by the parametric equations $$x = t ^ { 2 } \text { and } y = 2 t \quad - \sqrt { 2 } \leq t \leq \sqrt { 2 }$$ is shown in Figure 1 below. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{27339c29-c4a1-480c-b882-930f8dacc7af-13_1063_1022_607_507}
\end{figure} 8
  1. Find a Cartesian equation of the curve in the form \(y ^ { 2 } = \mathrm { f } ( x )\)
    8
  2. The point \(A\) lies on the curve where \(t = a\) The tangent to the curve at \(A\) is at an angle \(\theta\) to a line through \(A\) parallel to the \(x\)-axis. The point \(B\) has coordinates \(( 1,0 )\)
    The line \(A B\) is at an angle \(\phi\) to the \(x\)-axis.
    \includegraphics[max width=\textwidth, alt={}, center]{27339c29-c4a1-480c-b882-930f8dacc7af-14_846_936_678_552} 8
    1. By considering the gradient of the curve, show that $$\tan \theta = \frac { 1 } { a }$$ 8
  4. (ii) Find \(\tan \phi\) in terms of \(a\). 8
  5. (iii) Show that \(\tan 2 \theta = \tan \phi\)
AQA Paper 2 Specimen Q3
6 marks Moderate -0.3
3 A curve is defined by the parametric equations $$x = t ^ { 3 } + 2 , \quad y = t ^ { 2 } - 1$$ 3
  1. Find the gradient of the curve at the point where \(t = - 2\)
    [0pt] [4 marks]
    3
  2. Find a Cartesian equation of the curve.