4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 y + 20 = 0\).
- By completing the square, express this equation in the form
$$x ^ { 2 } + ( y - b ) ^ { 2 } = k$$
- Write down:
- the coordinates of \(C\);
- the radius of the circle, leaving your answer in surd form.
- A line has equation \(y = 2 x\).
- Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation \(x ^ { 2 } - 4 x + 4 = 0\).
- Hence show that the line is a tangent to the circle and find the coordinates of the point of contact, \(P\).
- Prove that the point \(Q ( - 1,4 )\) lies inside the circle.