Line-circle intersection points

A circle \(C\) has centre \(A\) and equation \(x^2 + y^2 - 4x - 6y = 3\).

1. Find the coordinates of \(A\) and the radius of \(C\). [3]

The line \(L\) with equation \(y = x + 5\) intersects \(C\) at the points \(P\) and \(Q\).

1. Determine the coordinates of \(P\) and \(Q\). [4]
2. The point \(B\) is on \(PQ\) and is such that \(AB\) is perpendicular to \(PQ\). Find the length of \(PB\). [2]
3. Show that the area of the smaller segment enclosed by \(C\) and \(L\) is \(4\pi - 8\). [2]

A circle has centre \(C(3, -8)\) and radius \(10\).

1. Express the equation of the circle in the form $$(x - a)^2 + (y - b)^2 = k$$ [2 marks]
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. [3 marks]
3. The line with equation \(y = 2x + 1\) intersects the circle.
   1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x^2 + 6x - 2 = 0$$ [3 marks]
   2. 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. [2 marks]