The distinct points \(P(2ap,\, ap^2)\) and \(Q(2aq,\, aq^2)\) lie on the curve \(x^2 = 4ay\), where \(a > 0\).
Given that
\[(p+q)^2 = p^2q^2 + 6pq + 5\,,\tag{\(*\)}\]
show that the line through \(P\) and \(Q\) is a tangent to the circle with centre \((0,\, 3a)\) and radius \(2a\).
Show that, for any given value of \(p\) with \(p^2 \neq 1\), there are two distinct real values of \(q\) that satisfy equation \((*)\).
Let these values be \(q_1\) and \(q_2\). Find expressions, in terms of \(p\), for \(q_1 + q_2\) and \(q_1 q_2\).
Show that, for any given value of \(p\) with \(p^2 \neq 1\), there is a triangle with one vertex at \(P\) such that all three vertices lie on the curve \(x^2 = 4ay\) and all three sides are tangents to the circle with centre \((0,\, 3a)\) and radius \(2a\).