10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-28_903_1010_219_539}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
Figure 5 shows a sketch of the quadratic curve \(C\) with equation
$$y = - \frac { 1 } { 4 } ( x + 2 ) ( x - b ) \quad \text { where } b \text { is a positive constant }$$
The line \(l _ { 1 }\) also shown in Figure 5,
- has gradient \(\frac { 1 } { 2 }\)
- intersects \(C\) on the negative \(x\)-axis and at the point \(P\)
- (i) Write down an equation for \(l _ { 1 }\)
(ii) Find, in terms of \(b\), the coordinates of \(P\)
Given that the line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and intersects \(C\) on the positive \(x\)-axis,
find, in terms of \(b\), an equation for \(l _ { 2 }\)
Given also that \(l _ { 2 }\) intersects \(C\) at the point \(P\)show that another equation for \(l _ { 2 }\) is
$$y = - 2 x + \frac { 5 b } { 2 } - 4$$Hence, or otherwise, find the value of \(b\)