6 The circle with centre \(C ( 5,8 )\) touches the \(y\)-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-5_485_631_370_715}
- Express the equation of the circle in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
- Verify that the point \(A ( 2,12 )\) lies on the circle.
- Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(s x + t y + u = 0\), where \(s , t\) and \(u\) are integers.
- The points \(P\) and \(Q\) lie on the circle, and the mid-point of \(P Q\) is \(M ( 7,12 )\).
- Show that the length of \(C M\) is \(n \sqrt { 5 }\), where \(n\) is an integer.
- Hence find the area of triangle \(P C Q\).