- The curve \(C\) with equation
$$y = \frac { p - 3 x } { ( 2 x - q ) ( x + 3 ) } \quad x \in \mathbb { R } , x \neq - 3 , x \neq 2$$
where \(p\) and \(q\) are constants, passes through the point \(\left( 3 , \frac { 1 } { 2 } \right)\) and has two vertical asymptotes
with equations \(x = 2\) and \(x = - 3\) with equations \(x = 2\) and \(x = - 3\)
- Explain why you can deduce that \(q = 4\)
- Show that \(p = 15\)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-38_616_889_842_587}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve \(C\). The region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 3\)
- Show that the exact value of the area of \(R\) is \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are rational constants to be found.