Parametric normal then bounded area

A question is this type if and only if it first requires finding the equation of a normal to a parametric curve at a given point, and then uses that normal line as one boundary to find the area of a region using parametric integration.

3 questions · Standard +0.8

1.07s Parametric and implicit differentiation
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Edexcel C34 2017 June Q14
16 marks Standard +0.8
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{29b56d51-120a-4275-a761-8b8aed7bca54-48_506_812_219_571} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 8 \cos ^ { 3 } \theta , \quad y = 6 \sin ^ { 2 } \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ Given that the point \(P\) lies on \(C\) and has parameter \(\theta = \frac { \pi } { 3 }\)
  1. find the coordinates of \(P\). The line \(l\) is the normal to \(C\) at \(P\).
  2. Show that an equation of \(l\) is \(y = x + 3.5\) The finite region \(S\), shown shaded in Figure 6, is bounded by the curve \(C\), the line \(l\), the \(y\)-axis and the \(x\)-axis.
  3. Show that the area of \(S\) is given by $$4 + 144 \int _ { 0 } ^ { \frac { \pi } { 3 } } \left( \sin \theta \cos ^ { 2 } \theta - \sin \theta \cos ^ { 4 } \theta \right) d \theta$$
  4. Hence, by integration, find the exact area of \(S\).
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
    END
Edexcel P4 2023 January Q8
11 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c46ca445-cf59-4664-931e-add9f2f81851-26_582_773_255_648} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} A curve \(C\) has parametric equations $$x = \sin ^ { 2 } t \quad y = 2 \tan t \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The point \(P\) with parameter \(t = \frac { \pi } { 4 }\) lies on \(C\).
The line \(l\) is the normal to \(C\) at \(P\), as shown in Figure 3.
  1. Show, using calculus, that an equation for \(l\) is $$8 y + 2 x = 17$$ The region \(S\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(x\)-axis.
  2. Find, using calculus, the exact area of \(S\).
Edexcel C4 2008 June Q8
16 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fb1924cc-9fa3-4fde-ba4d-6fb095f7f70b-11_639_972_228_484} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the curve \(C\) with parametric equations $$x = 8 \cos t , \quad y = 4 \sin 2 t , \quad 0 \leqslant t \leqslant \frac { \pi } { 2 } .$$ The point \(P\) lies on \(C\) and has coordinates \(( 4,2 \sqrt { } 3 )\).
  1. Find the value of \(t\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\).
  2. Show that an equation for \(l\) is \(y = - x \sqrt { 3 } + 6 \sqrt { 3 }\). The finite region \(R\) is enclosed by the curve \(C\), the \(x\)-axis and the line \(x = 4\), as shown shaded in Figure 3.
  3. Show that the area of \(R\) is given by the integral \(\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } 64 \sin ^ { 2 } t \cos t \mathrm {~d} t\).
  4. Use this integral to find the area of \(R\), giving your answer in the form \(a + b \sqrt { } 3\), where \(a\) and \(b\) are constants to be determined.