4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 12 y + 12 = 0\).
- By completing the square, express this equation in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
- Write down:
- the coordinates of \(C\);
- the radius of the circle.
- Show that the circle does not intersect the \(x\)-axis.
- The line with equation \(x + y = 4\) intersects the circle at the points \(P\) and \(Q\).
- Show that the \(x\)-coordinates of \(P\) and \(Q\) satisfy the equation
$$x ^ { 2 } + 3 x - 10 = 0$$
- Given that \(P\) has coordinates (2,2), find the coordinates of \(Q\).
- Hence find the coordinates of the midpoint of \(P Q\).