Verify parametric equations

A question is this type if and only if it asks to verify that given parametric equations satisfy a Cartesian equation or represent a specific curve type.

7 questions · Moderate -0.1

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Edexcel P4 2022 January Q2
4 marks Standard +0.3
2. The curve \(C\) has parametric equations $$x = \frac { t ^ { 4 } } { 2 t + 1 } \quad y = \frac { t ^ { 3 } } { 2 t + 1 } \quad t > 0$$
  1. Write down \(\frac { x } { y }\) in terms of \(t\), giving your answer in simplest form.
  2. Hence show that all points on \(C\) satisfy the equation $$x ^ { 3 } - 2 x y ^ { 3 } - y ^ { 4 } = 0$$
OCR MEI C4 2008 January Q8
18 marks Moderate -0.3
8 A curve has equation $$x ^ { 2 } + 4 y ^ { 2 } = k ^ { 2 } ,$$ where \(k\) is a positive constant.
  1. Verify that $$x = k \cos \theta , \quad y = \frac { 1 } { 2 } k \sin \theta ,$$ are parametric equations for the curve.
  2. Hence or otherwise show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x } { 4 y }\).
  3. Fig. 8 illustrates the curve for a particular value of \(k\). Write down this value of \(k\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9a8332ec-2216-4e1f-9768-ef175c9e159b-4_657_1071_938_577} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure}
  4. Copy Fig. 8 and on the same axes sketch the curves for \(k = 1 , k = 3\) and \(k = 4\). On a map, the curves represent the contours of a mountain. A stream flows down the mountain. Its path on the map is always at right angles to the contour it is crossing.
  5. Explain why the path of the stream is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 y } { x } .$$
  6. Solve this differential equation. Given that the path of the stream passes through the point \(( 2,1 )\), show that its equation is \(y = \frac { x ^ { 4 } } { 16 }\).
OCR MEI C4 Q2
4 marks Easy -1.2
2 Show that the curve, given by the parametric equations given below, represents a circle. $$x = 2 \cos \theta + 3 , y = 2 \sin \theta - 3$$ State the radius and centre of this circle.
OCR MEI C4 Q4
18 marks Moderate -0.3
4 A curve has equation $$x ^ { 2 } + 4 y ^ { 2 } = k ^ { 2 }$$ where \(k\) is a positive constant.
  1. Verify that $$x = k \cos \theta , \quad y = \frac { 1 } { 2 } k \sin \theta ,$$ are parametric equations for the curve.
  2. Hence or otherwise show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x } { 4 y }\).
  3. Fig. 8 illustrates the curve for a particular value of \(k\). Write down this value of \(k\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7278ce82-f710-44a7-945e-c194a4fb1744-4_666_1080_886_522} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure}
  4. Copy Fig. 8 and on the same axes sketch the curves for \(k = 1 , k = 3\) and \(k = 4\). On a map, the curves represent the contours of a mountain. A stream flows down the mountain. Its path on the map is always at right angles to the contour it is crossing.
  5. Explain why the path of the stream is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 y } { x } .$$
  6. Solve this differential equation. Given that the path of the stream passes through the point \(( 2,1 )\), show that its equation is \(y = \frac { x ^ { 4 } } { 16 }\).
OCR MEI C4 Q2
18 marks Moderate -0.3
2 A curve has equation $$x ^ { 2 } + 4 y ^ { 2 } = k ^ { 2 } ,$$ where \(k\) is a positive constant.
  1. Verify that $$x = k \cos \theta , \quad y = \frac { 1 } { 2 } k \sin \theta ,$$ are parametric equations for the curve.
  2. Hence or otherwise show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x } { 4 y }\).
  3. Fig. 8 illustrates the curve for a particular value of \(k\). Write down this value of \(k\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1601927c-74d7-4cc2-a7f2-2c2a2e8c2c4c-2_658_1070_861_567} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure}
  4. Copy Fig. 8 and on the same axes sketch the curves for \(k = 1 , k = 3\) and \(k = 4\). On a map, the curves represent the contours of a mountain. A stream flows down the mountain. Its path on the map is always at right angles to the contour it is crossing.
  5. Explain why the path of the stream is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 y } { x } .$$
  6. Solve this differential equation. Given that the path of the stream passes through the point \(( 2,1 )\), show that its equation is \(y = \frac { x ^ { 4 } } { 16 }\).
Edexcel Paper 1 2021 October Q13
3 marks Standard +0.3
  1. A curve \(C\) has parametric equations
$$x = \frac { t ^ { 2 } + 5 } { t ^ { 2 } + 1 } \quad y = \frac { 4 t } { t ^ { 2 } + 1 } \quad t \in \mathbb { R }$$ Show that all points on \(C\) satisfy $$( x - 3 ) ^ { 2 } + y ^ { 2 } = 4$$
OCR MEI Paper 3 2018 June Q8
8 marks Standard +0.8
8 A curve has parametric equations \(x = \frac { t } { 1 + t ^ { 3 } } , y = \frac { t ^ { 2 } } { 1 + t ^ { 3 } }\), where \(t \neq - 1\).
  1. In this question you must show detailed reasoning. Determine the gradient of the curve at the point where \(t = 1\).
  2. Verify that the cartesian equation of the curve is \(x ^ { 3 } + y ^ { 3 } = x y\).