2025 VJC P1 Q5

The function \(\mathrm{f}\) is given by

\[\mathrm{f}(x) = \begin{cases} 2 + \sqrt{2^2 - (x-2)^2}, & 0 < x \leq 4, \\ 2 - \sqrt{2^2 - (x-6)^2}, & 4 < x \leq 8. \end{cases}\]

It is given that \(\mathrm{f}(x) = \mathrm{f}(x+8)\) for all real values of \(x\).

1. On the diagram in the Printed Answer Book, sketch the graph of \(y = \mathrm{f}(x)\) for \(-6 \leq x \leq 7\), indicating clearly the coordinates of the end points and the points where the graph cuts the axes.[3]
2. Without integrating, write down the exact area of the region bounded by \(y = \mathrm{f}(x)\), the line \(x = 4\), the \(x\)-axis and the \(y\)-axis.[1]

The curve \(C\) has equation \(\frac{(y-3)^2}{a^2} - (x+2)^2 = 1\), where \(a\) is a positive real constant.

3. State the equations of the asymptotes of \(C\) in terms of \(a\).[1]
4. Determine the range of values of \(a\) if there is at most one intersection between \(C\) and the graph of \(y = \mathrm{f}(x)\).[2]