9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-24_889_666_258_703}
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\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of the curve \(C\) with equation
$$y = \frac { 1 } { 2 } x ^ { 2 } - 10 x + 22$$
- Write \(\frac { 1 } { 2 } x ^ { 2 } - 10 x + 22\) in the form
$$a ( x + b ) ^ { 2 } + c$$
where \(a , b\) and \(c\) are constants to be found.
The point \(M\) is the minimum turning point of \(C\), as shown in Figure 3.
- Deduce the coordinates of \(M\)
The line \(l\) is the normal to \(C\) at the point \(P\), as shown in Figure 3.
Given that \(l\) has equation \(y = k - \frac { 1 } { 8 } x\), where \(k\) is a constant, - find the coordinates of \(P\)
- find the value of \(k\)
Question 9 continues on the next page
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-25_903_682_299_605}
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\caption{Figure 4}
\end{figure}
Figure 4 is a copy of Figure 3. The finite region \(R\), shown shaded in Figure 4, is bounded by \(l , C\) and the line through \(M\) parallel to the \(y\)-axis.
- Identify the inequalities that define \(R\).