7 The circle \(S\) has centre \(C ( 8,13 )\) and touches the \(x\)-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-4_444_755_356_641}
- Write down an equation for \(S\), giving your answer in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
- The point \(P\) with coordinates \(( 3,1 )\) lies on the circle.
- Find the gradient of the straight line passing through \(P\) and \(C\).
- Hence find an equation of the tangent to the circle \(S\) at the point \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
- The point \(Q\) also lies on the circle \(S\), and the length of \(P Q\) is 10 . Calculate the shortest distance from \(C\) to the chord \(P Q\).