Let p and q be real parameters and consider the circle
x2+y2+2px+2qy=p2+q2.
(i) Show that the point A(0,q) lies on the circle.
(ii) Find the slope of the tangent to the circle at A.
(iii) The normal at A is required to pass through the origin. Establish the relation between p and q that must hold for this condition to be satisfied.
(iv) Subject to the relation obtained in part (iii), determine the equation of the normal at A.
(v) State the coordinates of the centre of the circle when the condition in part (iii) holds and comment on whether the centre lies on the normal obtained in part (iv).