10. The circle \(C\) has centre \(X ( 3,5 )\) and radius \(r\)
The line \(l\) has equation \(y = 2 x + k\), where \(k\) is a constant.
- Show that \(l\) and \(C\) intersect when
$$5 x ^ { 2 } + ( 4 k - 26 ) x + k ^ { 2 } - 10 k + 34 - r ^ { 2 } = 0$$
Given that \(l\) is a tangent to \(C\),
- show that \(5 r ^ { 2 } = ( k + p ) ^ { 2 }\), where \(p\) is a constant to be found.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{db4ec300-8081-4d29-acd5-0aae789d8f95-28_636_572_902_687}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
The line \(l\)
- cuts the \(y\)-axis at the point \(A\)
- touches the circle \(C\) at the point \(B\)
as shown in Figure 2.
Given that \(A B = 2 r\) - find the value of \(k\)