8 A circle has centre \(C ( 3 , - 8 )\) and radius 10 .
- Express the equation of the circle in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
- Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis.
- The tangent to the circle at the point \(A\) has gradient \(\frac { 5 } { 2 }\). Find an equation of the line \(C A\), giving your answer in the form \(r x + s y + t = 0\), where \(r , s\) and \(t\) are integers.
- The line with equation \(y = 2 x + 1\) intersects the circle.
- Show that the \(x\)-coordinates of the points of intersection satisfy the equation
$$x ^ { 2 } + 6 x - 2 = 0$$
- Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.